Chapter 6 Resource Masters
Consumable Workbooks Many of the worksheets contained in the Chapter Resource Masters are available as consumable workbooks in both English and Spanish.
ISBN10 ISBN13
Study Guide and Intervention Workbook 0-07-890848-5 978-0-07-890848-4
Homework Practice Workbook 0-07-890849-3 978-0-07-890849-1
Spanish Version
Homework Practice Workbook 0-07-890853-1 978-0-07-890853-8
Answers for Workbooks The answers for Chapter 6 of these workbooks can be found in the back of this Chapter Resource Masters booklet.
StudentWorks PlusTM This CD-ROM includes the entire Student Edition test along with the English workbooks listed above.
TeacherWorks PlusTM All of the materials found in this booklet are included for viewing, printing, and editing in this CD-ROM.
Spanish Assessment Masters (ISBN10: 0-07-890856-6, ISBN13: 978-0-07-890856-9)These masters contain a Spanish version of Chapter 6 Test Form 2A and Form 2C.
Copyright © by the McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe Geometry program. Any other reproduction, for sale or other use, is expressly prohibited.
Send all inquiries to:
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, OH 43240 - 4027
ISBN13: 978-0-07-890515-5
ISBN10: 0-07-890515-X
Printed in the United States of America.
1 2 3 4 5 6 7 8 9 10 024 14 13 12 11 10 09 08
iii
Teacher’s Guide to Using the Chapter 6
Resource Masters .........................................iv
Chapter ResourcesStudent-Built Glossary ....................................... 1
Anticipation Guide (English) .............................. 3
Anticipation Guide (Spanish) ............................. 4
Lesson 6-1Angles of Polygons
Study Guide and Intervention ............................ 5
Skills Practice .................................................... 7
Practice .............................................................. 8
Word Problem Practice ..................................... 9
Enrichment ...................................................... 10
Lesson 6-2Parallelograms
Study Guide and Intervention .......................... 11
Skills Practice .................................................. 13
Practice ............................................................ 14
Word Problem Practice ................................... 15
Enrichment ...................................................... 16
Lesson 6-3Tests for Parallelograms
Study Guide and Intervention .......................... 17
Skills Practice .................................................. 19
Practice ............................................................ 20
Word Problem Practice ................................... 21
Enrichment ...................................................... 22
Lesson 6-4Rectangles
Study Guide and Intervention .......................... 23
Skills Practice .................................................. 25
Practice ............................................................ 26
Word Problem Practice ................................... 27
Enrichment ...................................................... 28
TI-Nspire Activity ............................................. 29
Geometer’s Sketchpad Activity ....................... 30
Lesson 6-5Rhombi and Squares
Study Guide and Intervention .......................... 31
Skills Practice .................................................. 33
Practice ............................................................ 34
Word Problem Practice ................................... 35
Enrichment ...................................................... 36
Lesson 6-6Trapezoids and Kites
Study Guide and Intervention .......................... 37
Skills Practice .................................................. 39
Practice ............................................................ 40
Word Problem Practice ................................... 41
Enrichment ...................................................... 42
AssessmentStudent Recording Sheet ................................ 43
Rubric for Extended Response ....................... 44
Chapter 6 Quizzes 1 and 2 ............................. 45
Chapter 6 Quizzes 3 and 4 ............................. 46
Chapter 6 Mid-Chapter Test ............................ 47
Chapter 6 Vocabulary Test ............................. 48
Chapter 6 Test, Form 1 ................................... 49
Chapter 6 Test, Form 2A ................................. 51
Chapter 6 Test, Form 2B ................................. 53
Chapter 6 Test, Form 2C ................................ 55
Chapter 6 Test, Form 2D ................................ 57
Chapter 6 Test, Form 3 ................................... 59
Chapter 6 Extended-Response Test ............... 61
Standardized Test Practice ............................. 62
Unit 2 Test ....................................................... 65
Answers ......................................... A24–A34
ContentsC
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Teacher’s Guide to Using the
Chapter 6 Resource MastersThe Chapter 6 Resource Masters includes the core materials needed for Chapter 6. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing on the
TeacherWorks PlusTM CD-ROM.
Chapter Resources
Student-Built Glossary (pages 1–2) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to recording definitions and/or examples for each term. You may suggest that student highlight or star the terms with which they are not familiar. Give to students before beginning Lesson 6-1. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson.
Anticipation Guide (pages 3–4) This mas-ter presented in both English and Spanish is a survey used before beginning the chapter to pinpoint what students may or may not know about the concepts in the chapter. Students will revisit this survey after they complete the chapter to see if their percep-tions have changed.
Lesson Resources
Study Guide and Intervention These masters provide vocabulary, key concepts, additional worked-out examples and Check Your Progress exercises to use as a reteaching activity. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent.
Skills Practice This master focuses more on the computational nature of the lesson. Use as an additional practice option or as homework for second-day teaching of the lesson.
Practice This master closely follows the types of problems found in the Exercises sec-tion of the Student Edition and includes word problems. Use as an additional prac-tice option or as homework for second-day teaching of the lesson.
Word Problem Practice This master includes additional practice in solving word problems that apply the concepts of the lesson. Use as an additional practice or as homework for second-day teaching of the lesson.
Enrichment These activities may extend the concepts of the lesson, offer a historical or multicultural look at the concepts, or widen students’ perspectives on the math-ematics they are learning. They are written for use with all levels of students.
Graphing Calculator or Spreadsheet
Activities These activities present ways in which technology can be used with the con-cepts in some lessons of this chapter. Use as an alternative approach to some concepts or as an integral part of your lesson presenta-tion.
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Assessment Options
The assessment masters in the Chapter 6
Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assess-ment.
Student Recording Sheet This master corresponds with the standardized test practice at the end of the chapter.
Extended-Response Rubric This master provides information for teachers and students on how to assess performance on open-ended questions.
Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter.
Mid-Chapter Test This 1-page test pro-vides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions.
Vocabulary Test This test is suitable for all students. It includes a list of vocabulary words and 10 questions to assess students’ knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests.
Leveled Chapter Tests
• Form 1 contains multiple-choice questions and is intended for use with below grade level students.
• Forms 2A and 2B contain multiple-choice questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.
• Forms 2C and 2D contain free-response questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.
• Form 3 is a free-response test for use with above grade level students.
All of the above mentioned tests include a free-response Bonus question.
Extended-Response Test Performance assessment tasks are suitable for all students. Samples answers and a scoring rubric are included for evaluation.
Standardized Test Practice These three pages are cumulative in nature. It includes three parts: multiple-choice questions with bubble-in answer format, griddable questions with answer grids, and short-answer free-response questions.
Answers• The answers for the Anticipation Guide
and Lesson Resources are provided as reduced pages.
• Full-size answer keys are provided for the assessment masters.
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NAME DATE PERIOD
Chapter 6 1 Glencoe Geometry
Student-Built Glossary
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 6.
As you study the chapter, complete each term’s definition or description.
Remember to add the page number where you found the term. Add these pages to
your Geometry Study Notebook to review vocabulary at the end of the chapter.
6
Vocabulary TermFound
on PageDefinition/Description/Example
base
diagonal
isosceles trapezoid
legs
midsegment of a trapezoid
(continued on the next page)
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Chapter 6 2 Glencoe Geometry
Student-Built Glossary (continued)6
Vocabulary TermFound
on PageDefinition/Description/Example
parallelogram
rectangle
rhombus
square
trapezoid
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Chapter 6 3 Glencoe Geometry
Anticipation Guide
Quadrilaterals
Before you begin Chapter 6
• Read each statement.
• Decide whether you Agree (A) or Disagree (D) with the statement.
• Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).
After you complete Chapter 6
• Reread each statement and complete the last column by entering an A or a D.
• Did any of your opinions about the statements change from the first column?
• For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.
6
STEP 1A, D, or NS
StatementSTEP 2A or D
1. A triangle has no diagonals.
2. A diagonal of a polygon is a segment joining the midpoints of two sides of the polygon.
3. The sum of the measures of the angles in a polygon can be determined by subtracting 2 from the number of sides and multiplying the result by 180.
4. For a quadrilateral to be a parallelogram it must have two pairs of parallel sides.
5. The diagonals of a parallelogram are congruent.
6. If you know that one pair of opposite sides of a quadrilateral is both parallel and congruent, then you know the quadrilateral is a parallelogram.
7. If a quadrilateral is a rectangle, then all four angles are congruent.
8. The diagonals of a rhombus are congruent.
9. The properties of a rhombus are not true for a square.
10. A trapezoid has only one pair of parallel sides.
11. The median of a trapezoid is perpendicular to the bases.
12. An isosceles trapezoid has exactly one pair of congruent sides.
Step 2
Step 1
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NOMBRE FECHA PERÍODO
Antes de comenzar el Capítulo 6
• Lee cada enunciado.
• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.
• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta, escribe NS (No estoy seguro(a).
Después de completar el Capítulo 6
• Vuelve a leer cada enunciado y completa la última columna con una A o una D.
• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?
• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con los enunciados que marcaste con una D.
Ejercicios preparatorios
Cuadriláteros
6
Paso 1
PASO 1A, D o NS
EnunciadoPASO 2A o D
1. Un triángulo no tiene diagonales.
2. La diagonal de un polígono es un segmento que une los puntos medios de dos lados del polígono.
3. La suma de las medidas de los ángulos de un polígono puede determinarse restando 2 del número de lados y multiplicando el resultado por 180.
4. Para que un cuadrilátero sea un paralelogramo debe tener dos pares de lados paralelos.
5. Las diagonales de un paralelogramo son congruentes.
6. Si sabes que un par de lados opuestos de un cuadrilátero son paralelos y congruentes, entonces sabes que el cuadrilátero es un paralelogramo.
7. Si un cuadrilátero es un rectángulo, entonces todos los cuatro lados son congruentes.
8. Las diagonales de un rombo son congruentes.
9. Las propiedades de un rombo no son verdaderas para un cuadrado.
10. Un trapecio sólo tiene dos lados paralelos.
11. La mediana de un trapecio es perpendicular a las bases.
12. Un trapecio isósceles tiene exactamente un par de lados congruentes.
Paso 2
Capítulo 6 4 Geometría de Glencoe
Less
on
6-1
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Chapter 6 5 Glencoe Geometry
Study Guide and Intervention
Angles of Polygons
Polygon Interior Angles Sum The segments that connect the nonconsecutive vertices of a polygon are called diagonals. Drawing all of the diagonals from one vertex of an n-gon separates the polygon into n - 2 triangles. The sum of the measures of the interior angles of the polygon can be found by adding the measures of the interior angles of those n - 2 triangles.
A convex polygon has
13 sides. Find the sum of the measures
of the interior angles.
(n - 2) 180 = (13 - 2) 180
= 11 180
= 1980
The measure of an
interior angle of a regular polygon is
120. Find the number of sides.
The number of sides is n, so the sum of the measures of the interior angles is 120n.
120n = (n - 2) 180
120n = 180n - 360
-60n = -360
n = 6
Exercises
Find the sum of the measures of the interior angles of each convex polygon.
1. decagon 2. 16-gon 3. 30-gon
4. octagon 5. 12-gon 6. 35-gon
The measure of an interior angle of a regular polygon is given. Find the number
of sides in the polygon.
7. 150 8. 160 9. 175
10. 165 11. 144 12. 135
13. Find the value of x.
6 -1
7x°
(4x + 10)°
(4x + 5)°
(6x + 10)°
(5x - 5)°
A B
C
D
E
Polygon Interior Angle
Sum TheoremThe sum of the interior angle measures of an n-sided convex polygon is (n - 2) · 180.
Example 1 Example 2
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Chapter 6 6 Glencoe Geometry
Study Guide and Intervention (continued)
Angles of Polygons
Polygon Exterior Angles Sum There is a simple relationship among the exterior angles of a convex polygon.
Find the sum of the measures of the exterior angles, one at
each vertex, of a convex 27-gon.
For any convex polygon, the sum of the measures of its exterior angles, one at each
vertex, is 360.
Find the measure of each exterior angle of
regular hexagon ABCDEF.
The sum of the measures of the exterior angles is 360 and a hexagon
has 6 angles. If n is the measure of each exterior angle, then
6n = 360
n = 60
The measure of each exterior angle of a regular hexagon is 60.
Exercises
Find the sum of the measures of the exterior angles of each convex polygon.
1. decagon 2. 16-gon 3. 36-gon
Find the measure of each exterior angle for each regular polygon.
4. 12-gon 5. hexagon 6. 20-gon
7. 40-gon 8. heptagon 9. 12-gon
10. 24-gon 11. dodecagon 12. octagon
AB
C
DE
F
6 -1
Polygon Exterior Angle
Sum Theorem
The sum of the exterior angle measures of a convex polygon, one angle
at each vertex, is 360.
Example 1
Example 2
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Lesso
n 6
-1
Chapter 6 7 Glencoe Geometry
Skills Practice
Angles of Polygons
Find the sum of the measures of the interior angles of each convex polygon.
1. nonagon 2. heptagon 3. decagon
The measure of an interior angle of a regular polygon is given. Find the
number of sides in the polygon.
4. 108 5. 120 6. 150
Find the measure of each interior angle.
7. 8.
9. 10.
Find the measures of each interior angle of each regular polygon.
11. quadrilateral 12. pentagon 13. dodecagon
Find the measures of each exterior angle of each regular polygon.
14. octagon 15. nonagon 16. 12-gon
6-1
x°
x°
(2x - 15)°
(2x - 15)°
A B
CD
(2x)°
(2x + 20)° (3x - 10)°
(2x - 10)°
L M
NP
(2x + 16)°
(x + 14)° (x + 14)°
(2x + 16)° (7x)° (7x)°
(4x)° (4x)°
(7x)° (7x)°
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Chapter 6 8 Glencoe Geometry
Find the sum of the measures of the interior angles of each convex polygon.
1. 11-gon 2. 14-gon 3. 17-gon
The measure of an interior angle of a regular polygon is given. Find the
number of sides in the polygon.
4. 144 5. 156 6. 160
Find the measure of each interior angle.
7. 8.
Find the measures of an exterior angle and an interior angle given the number of
sides of each regular polygon. Round to the nearest tenth, if necessary.
9. 16 10. 24 11. 30
12. 14 13. 22 14. 40
15. CRYSTALLOGRAPHY Crystals are classified according to seven crystal systems. The
basis of the classification is the shapes of the faces of the crystal. Turquoise belongs to
the triclinic system. Each of the six faces of turquoise is in the shape of a parallelogram.
Find the sum of the measures of the interior angles of one such face.
Practice
Angles of Polygons
6 -1
x°
(2x + 15)° (3x - 20)°
(x + 15)°
J K
MN
(6x - 4)°
(2x + 8)° (6x - 4)°
(2x + 8)°
Lesso
n 6
-1
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Chapter 6 9 Glencoe Geometry
Word Problem Practice
Angles of Polygons
La Tribuna
1. ARCHITECTURE In the Uffizi gallery in Florence, Italy, there is a room built by Buontalenti called the Tribune (La
Tribuna in Italian). This room is shaped like a regular octagon.
What angle do consecutive walls of the Tribune make with each other?
2. BOXES Jasmine is designing boxes she will use to ship her jewelry. She wants to shape the box like a regular polygon. In order for the boxes to pack tightly, she decides to use a regular polygon that has the property that the measure of its interior angles is half the measure of its exterior angles. What regular polygon should she use?
3. THEATER A theater floor plan is shown in the figure. The upper five sides are part of a regular dodecagon.
Find m∠1.
4. ARCHEOLOGY Archeologists unearthed parts of two adjacent walls of an ancient castle.
Before it was unearthed, they knew from ancient texts that the castle was shaped like a regular polygon, but nobody knew how many sides it had. Some said 6, others 8, and some even said 100. From the information in the figure, how many sides did the castle really have?
5. POLYGON PATH In Ms. Rickets’ math class, students made a “polygon path” that consists of regular polygons of 3, 4, 5, and 6 sides joined together as shown.
a. Find m∠2 and m∠5.
b. Find m∠3 and m∠4.
c. What is m∠1?
6 -1
2
1
3
4
5
stage
1
24˚
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Chapter 6 10 Glencoe Geometry
Enrichment
Central Angles of Regular PolygonsYou have learned about the interior and exterior angles of a polygon. Regular polygons also have central angles. A central angle is measured from the center of the polygon.
The center of a polygon is the point equidistant from all of the vertices of the polygon, just as the center of a circle is the point equidistant from all of the points on the circle. The central angle is the angle drawn with the vertex at the center of the circle and the sides of angle drawn through consecutive vertices of the polygon. One of the central angles that can be drawn in this regular hexagon is ∠APB. You may remember from making circle graphs that there are 360° around the center of a circle.
1. By using logic or by drawing sketches, find the measure of the central angle of each regular polygon.
2. Make a conjecture about how the measure of a central angle of a regular polygon relates to the measures of the interior angles and exterior angles of a regular polygon.
3. CHALLENGE In obtuse △ABC, −−−
BC is the longest side. −−
AC is also a side of a 21-sided regular polygon.
−− AB is also a side of a 28-sided regular polygon. The
21-sided regular polygon and the 28-sided regular polygon have the same center point P. Find n if
−−− BC is a side of a n-sided regular polygon that has
center point P.
(Hint: Sketch a circle with center P and place points A, B, and C on the circle.)
6 -1
A
B
P
r
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Chapter 6 11 Glencoe Geometry
Study Guide and Intervention
Parallelograms
Sides and Angles of Parallelograms A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Here are four important properties of parallelograms.
If ABCD is a parallelogram, find the value of each variable.−−
AB and −−−
CD are opposite sides, so −−
AB " −−−
CD .
2a = 34
a = 17
∠A and ∠C are opposite angles, so ∠A " ∠C.
8b = 112
b = 14
Exercises
Find the value of each variable.
1. 2.
3. 4.
5. 6.
P Q
RS
BA
D C34
2a
8b°
112°
4y°
3x° 8y
88
6x°
72x
30x 150
2y
6 -2
If PQRS is a parallelogram, then
If a quadrilateral is a parallelogram,
then its opposite sides are congruent.
−−−
PQ " −−
SR and −−
PS " −−−
QR
If a quadrilateral is a parallelogram,
then its opposite angles are congruent.
∠P " ∠R and ∠S " ∠Q
If a quadrilateral is a parallelogram,
then its consecutive angles are
supplementary.
∠P and ∠S are supplementary; ∠S and ∠R are supplementary;
∠R and ∠Q are supplementary; ∠Q and ∠P are supplementary.
If a parallelogram has one right angle,
then it has four right angles.
If m∠P = 90, then m∠Q = 90, m∠R = 90, and m∠S = 90.
Example
3y
12
6x
60°
55° 5x°
2y°
6x° 12x°
3y°
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Chapter 6 12 Glencoe Geometry
Study Guide and Intervention (continued)
Parallelograms
Diagonals of Parallelograms Two important properties of parallelograms deal with their diagonals.
Find the value of x and y in parallelogram ABCD.
The diagonals bisect each other, so AE = CE and DE = BE.
6x = 24 4y = 18
x = 4 y = 4.5
Exercises
Find the value of each variable.
1. 2. 3.
4. 5. 6.
COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of ABCD with the given vertices.
7. A(3, 6), B(5, 8), C(3, −2), and D(1, −4) 8. A(−4, 3), B(2, 3), C(−1, −2), and D(−7, −2)
9. PROOF Write a paragraph proof of the following.
Given: !ABCD
Prove: △AED # △BEC
A B
CD
P
A B
E
CD
6x
18
24
4y
3x
4y
812
4x
2y
28
3x
30°10
y
A B
CDE
2x°
60° 4y°
3x°2y
12
6 -2
If ABCD is a parallelogram, then
If a quadrilateral is a parallelogram, then
its diagonals bisect each other.
AP = PC and DP = PB
If a quadrilateral is a parallelogram, then each
diagonal separates the parallelogram
into two congruent triangles.
△ACD # △CAB and △ADB # △CBD
Example
x
17
4y
Lesso
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-2
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Chapter 6 13 Glencoe Geometry
Skills Practice
Parallelograms
ALGEBRA Find the value of each variable.
1. 2.
3. 4.
5. 6.
COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals
of HJKL with the given vertices.
7. H(1, 1), J(2, 3), K(6, 3), L(5, 1) 8. H(-1, 4), J(3, 3), K(3, -2), L(-1, -1)
9. PROOF Write a paragraph proof of the theorem Consecutive angles in a parallelogram
are supplementary.
6-2
x-2 y+ 1
1926
x° 44°
y°
4x-
2
3y-
1
y+5 x
+10
2b-1 b+ 3
3a- 5
2a
a+ 146a+ 4
10b+ 1
9b+ 8
(x+ 8)°
(3x)°
(y+ 9)°
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Chapter 6 14 Glencoe Geometry
Practice
Parallelograms
ALGEBRA Find the value of each variable.
1. 2.
3. 4.
ALGEBRA Use RSTU to find each measure or value.
5. m∠RST = 6. m∠STU =
7. m∠TUR = 8. b =
COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals
of PRYZ with the given vertices.
9. P(2, 5), R(3, 3), Y(-2, -3), Z(-3, -1) 10. P(2, 3), R(1, -2), Y(-5, -7), Z(-4, -2)
11. PROOF Write a paragraph proof of the following.
Given: !PRST and !PQVU
Prove: ∠V " ∠S
12. CONSTRUCTION Mr. Rodriquez used the parallelogram at the right to design a herringbone pattern for a paving stone. He will use the paving stone for a sidewalk. If m∠1 is 130, find m∠2, m∠3, and m∠4.
TU
SR
B30° 23
4b - 1
25°
P R
S
U V
Q
T
1 2
34
6 -2
(4x)°
(y+10)°
(2y-40)°
4x
5y-8
3y+1
x+
6
y+3 15
x-3
12
b+1
2b
a+2
3a-4
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Chapter 6 15 Glencoe Geometry
1. WALKWAY A walkway is made by
adjoining four parallelograms as shown.
Are the end segments a and e parallel to
each other? Explain.
2. DISTANCE Four friends live at the four
corners of a block shaped like a
parallelogram. Gracie lives 3 miles away
from Kenny. How far apart do Teresa
and Travis live from each other?
3. SOCCER Four soccer players are located
at the corners of a parallelogram. Two
of the players in opposite corners are
the goalies. In order for goalie A to be
able to see the three others, she must
be able to see a certain angle x in her
field of vision.
What angle does the other goalie have to
be able to see in order to keep an eye on
the other three players?
4. VENN DIAGRAMS Make a Venn
diagram showing the relationship
between squares, rectangles, and
parallelograms.
5. SKYSCRAPERS On vacation, Tony’s
family took a helicopter tour of the city.
The pilot said the
newest building
in the city was
the building with
this top view. He
told Tony that the
exterior angle by
the front entrance
is 72°. Tony
wanted to know
more about the building, so he drew this
diagram and used his geometry skills to
learn a few more things. The front
entrance is next to vertex B.
a. What are the measures of the four
angles of the parallelogram?
b. How many pairs of congruent
triangles are there in the figure?
What are they?
Word Problem Practice
Parallelograms
A B
C D
E
72˚
Goalie BPlayer A
Goalie A
x
Player B
TeresaTravis
Gracie Kenny
a
e
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Chapter 6 16 Glencoe Geometry
Enrichment
Diagonals of ParallelogramsIn some drawings the diagonal of a parallelogram appears to be the angle bisector of both
opposite angles. When might that be true?
1. Given: Parallelogram PQRS with diagonal −−
PR . −−
PR is an angle bisector of ∠QPS and ∠QRS.
What type of parallelogram is PQRS? Justify
your answer.
2. Given: Parallelogram WPRK with angle
bisector −−−
KD , DP = 5, and WD = 7.
Find WK and KR.
3. Refer to Exercise 2. Write a statement
about parallelogram WPRK and angle
bisector −−−
KD .
4. Given: Parallelogram ABCD with
diagonal −−−
BD and angle bisector −−
BP .
PD = 5, BP = 6, and CP = 6.
The perimeter of triangle PCD is 15.
Find AB and BC.
P
Q R
S
PD
A
BC
PD
R
K
W
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Less
on
6-3
Chapter 6 17 Glencoe Geometry
Study Guide and Intervention
Tests for Parallelograms
Conditions for Parallelograms There are many ways to establish that a quadrilateral is a parallelogram.
Find x and y so that FGHJ is a
parallelogram.
FGHJ is a parallelogram if the lengths of the opposite sides are equal.
6x + 3 = 15 4x - 2y = 2
6x = 12 4(2) - 2y = 2
x = 2 8 - 2y = 2
-2y = -6
y = 3
Exercises
Find x and y so that the quadrilateral is a parallelogram.
1. 2.
3. 4.
5. 6.
A B
CD
E
J H
GF 6x + 3
15
4x - 2y 2
2x - 2
2y 8
12 11x°
5y° 55°
5x°5y°
25°
(x + y)°
2x°
30°
24°
6y°
3x°
9x°6y
45°18
6 -3
If: If:
both pairs of opposite sides are parallel, −−
AB || −−−
DC and −−
AD || −−−
BC ,
both pairs of opposite sides are congruent, −−
AB " −−−
DC and −−
AD " −−−
BC ,
both pairs of opposite angles are congruent, ∠ABC " ∠ADC and ∠DAB " ∠BCD,
the diagonals bisect each other, −−
AE " −−
CE and −−
DE " −−
BE ,
one pair of opposite sides is congruent and parallel, −−
AB || −−−
CD and −−
AB " −−−
CD , or −−
AD || −−−
BC and −−
AD " −−−
BC ,
then: the figure is a parallelogram. then: ABCD is a parallelogram.
Example
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Chapter 6 18 Glencoe Geometry
Study Guide and Intervention (continued)
Tests for Parallelograms
Parallelograms on the Coordinate Plane On the coordinate plane, the Distance, Slope, and Midpoint Formulas can be used to test if a quadrilateral is a parallelogram.
Determine whether ABCD is a parallelogram.
The vertices are A(-2, 3), B(3, 2), C(2, -1), and D(-3, 0).
Method 1: Use the Slope Formula, m = y
2 - y
1 − x
2 - x
1
.
slope of −−−
AD = 3 - 0 −
-2 - (-3) =
3 −
1 = 3 slope of
−−− BC =
2 - (-1) −
3 - 2 =
3 −
1 = 3
slope of −−
AB = 2 - 3 −
3 - (-2) = - 1 −
5 slope of
−−− CD = -1 - 0
− 2 - (-3)
= - 1 − 5
Since opposite sides have the same slope, −−
AB || −−−
CD and −−−
AD || −−−
BC . Therefore, ABCD is a parallelogram by definition.
Method 2: Use the Distance Formula, d = √ ######### (x2 - x
1)2 + ( y
2 - y
1)2 .
AB = √ ######### (-2 - 3)2 + (3 - 2)2 = √ ### 25 + 1 or √ ## 26
CD = √ ########## (2 - (-3))2 + (-1 - 0)2 = √ ### 25 + 1 or √ ## 26
AD = √ ########## (-2 - (-3))2 + (3 - 0)2 = √ ### 1 + 9 or √ ## 10
BC = √ ######### (3 - 2)2 + (2 - (-1))2 = √ ### 1 + 9 or √ ## 10
Since both pairs of opposite sides have the same length, −−
AB $ −−−
CD and −−−
AD $ −−−
BC . Therefore, ABCD is a parallelogram by Theorem 6.9.
Exercises
Graph each quadrilateral with the given vertices. Determine whether the figure is
a parallelogram. Justify your answer with the method indicated.
1. A(0, 0), B(1, 3), C(5, 3), D(4, 0); 2. D(-1, 1), E(2, 4), F(6, 4), G(3, 1);Slope Formula Slope Formula
3. R(-1, 0), S(3, 0), T(2, -3), U(-3, -2); 4. A(-3, 2), B(-1, 4), C(2, 1), D(0, -1);Distance Formula Distance and Slope Formulas
5. S(-2, 4), T(-1, -1), U(3, -4), V(2, 1); 6. F(3, 3), G(1, 2), H(-3, 1), I(-1, 4);Distance and Slope Formulas Midpoint Formula
7. A parallelogram has vertices R(-2, -1), S(2, 1), and T(0, -3). Find all possible coordinates for the fourth vertex.
x
y
O
C
A
D
B
6 -3
Example
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Lesso
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-3
Chapter 6 19 Glencoe Geometry
Skills Practice
Tests for Parallelograms
Determine whether each quadrilateral is a parallelogram. Justify your answer.
1. 2.
3. 4.
COORDINATE GEOMETRY Graph each quadrilateral with the given vertices.
Determine whether the figure is a parallelogram. Justify your answer with the
method indicated.
5. P(0, 0), Q(3, 4), S(7, 4), Y(4, 0); Slope Formula
6. S(-2, 1), R(1, 3), T(2, 0), Z(-1, -2); Distance and Slope Formulas
7. W(2, 5), R(3, 3), Y(-2, -3), N(-3, 1); Midpoint Formula
ALGEBRA Find x and y so that each quadrilateral is a parallelogram.
8. 9.
10. 11.
x + 16
y + 192y
2x - 8
2y - 11
y + 3
4x - 3
3x
(4x - 35)°
(3x + 10)°(2y - 5)°
(y + 15)°
3x - 14
x + 20
3y + 2y + 20
6 -3
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Chapter 6 20 Glencoe Geometry
Determine whether each quadrilateral is a parallelogram. Justify your answer.
1. 2.
3. 4.
COORDINATE GEOMETRY Graph each quadrilateral with the given vertices.
Determine whether the figure is a parallelogram. Justify your answer with the
method indicated.
5. P(-5, 1), S(-2, 2), F(-1, -3), T(2, -2); Slope Formula
6. R(-2, 5), O(1, 3), M(-3, -4), Y(-6, -2); Distance and Slope Formulas
ALGEBRA Find x and y so that the quadrilateral is a parallelogram.
7. 8.
9. 10.
11. TILE DESIGN The pattern shown in the figure is to consist of congruent parallelograms. How can the designer be certain that the shapes are parallelograms?
118° 62°
118°62°
(5x + 29)°
(7x - 11)°(3y + 15)°
(5y - 9)°
3y - 5
-3x + 4
-4x - 22y + 8
-4x + 6
-6x
7y + 312y - 7
-4y - 2
x + 12
-2x + 6
y + 23
Practice
Tests for Parallelograms
6 -3
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Chapter 6 21 Glencoe Geometry
1. BALANCING Nikia, Madison, Angela,
and Shelby are balancing themselves
on an “X”-shaped floating object. To
balance themselves, they want to
make themselves the vertices of a
parallelogram.
In order to achieve this, do all four of
them have to be the same distance from
the center of the object? Explain.
2. COMPASSES Two compass needles
placed side by side on a table are both
2 inches long and point due north. Do
they form the sides of a parallelogram?
3. FORMATION Four jets are flying in
formation. Three of the jets are shown
in the graph. If the four jets are located
at the vertices of a parallelogram, what
are the three possible locations of the
missing jet?
4. STREET LAMPS When a coordinate
plane is placed over the Harrisville town
map, the four street lamps in the center
are located as shown. Do the four lamps
form the vertices of a parallelogram?
Explain.
5. PICTURE FRAME Aaron is making a
wooden picture frame in the shape of a
parallelogram. He has two pieces of
wood that are 3 feet long and two that
are 4 feet long.
a. If he connects the pieces of wood at
their ends to each other, in what
order must he connect them to make
a parallelogram?
b. How many different parallelograms
could he make with these four
lengths of wood?
c. Explain something Aaron might do
to specify precisely the shape of the
parallelogram.
5
5y
x
O–5
–5
5
5
y
xO
Shelby
Madison
Nikia
Angela
6 -3 Word Problem Practice
Tests for Parallelograms
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Chapter 6 22 Glencoe Geometry
Enrichment
Tests for Parallelograms
By definition, a quadrilateral is a parallelogram if and only if both pairs of opposite sides
are parallel. What conditions other than both pairs of opposite sides parallel will guarantee
that a quadrilateral is a parallelogram? In this activity, several possibilities will be
investigated by drawing quadrilaterals to satisfy certain conditions. Remember that any
test that seems to work is not guaranteed to work unless it can be formally proven.
Complete.
1. Draw a quadrilateral with one pair of opposite sides congruent.
Must it be a parallelogram?
2. Draw a quadrilateral with both pairs of opposite sides congruent.
Must it be a parallelogram?
3. Draw a quadrilateral with one pair of opposite sides parallel and
the other pair of opposite sides congruent. Must it be a
parallelogram?
4. Draw a quadrilateral with one pair of opposite sides both parallel
and congruent. Must it be a parallelogram?
5. Draw a quadrilateral with one pair of opposite angles congruent.
Must it be a parallelogram?
6. Draw a quadrilateral with both pairs of opposite angles congruent.
Must it be a parallelogram?
7. Draw a quadrilateral with one pair of opposite sides parallel and
one pair of opposite angles congruent. Must it be a parallelogram?
6 -3
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Chapter 6 23 Glencoe Geometry
Study Guide and Intervention
Rectangles
Properties of Rectangles A rectangle is a quadrilateral with four right angles. Here are the properties of rectangles.
A rectangle has all the properties of a parallelogram.
• Opposite sides are parallel.
• Opposite angles are congruent.
• Opposite sides are congruent.
• Consecutive angles are supplementary.
• The diagonals bisect each other.
Also:
• All four angles are right angles. ∠UTS, ∠TSR, ∠SRU, and ∠RUT are right angles.
• The diagonals are congruent. −−
TR # −−−
US
T S
RU
Q
Quadrilateral RUTS
above is a rectangle. If US = 6x + 3 and
RT = 7x - 2, find x.
The diagonals of a rectangle are congruent, so US = RT.
6x + 3 = 7x - 2
3 = x - 2
5 = x
Quadrilateral RUTS
above is a rectangle. If m∠STR = 8x + 3
and m∠UTR = 16x - 9, find m∠STR.
∠UTS is a right angle, so m∠STR + m∠UTR = 90.
8x + 3 + 16x - 9 = 90
24x - 6 = 90
24x = 96
x = 4
m∠STR = 8x + 3 = 8(4) + 3 or 35
Exercises
Quadrilateral ABCD is a rectangle.
1. If AE = 36 and CE = 2x - 4, find x.
2. If BE = 6y + 2 and CE = 4y + 6, find y.
3. If BC = 24 and AD = 5y - 1, find y.
4. If m∠BEA = 62, find m∠BAC.
5. If m∠AED = 12x and m∠BEC = 10x + 20, find m∠AED.
6. If BD = 8y - 4 and AC = 7y + 3, find BD.
7. If m∠DBC = 10x and m∠ACB = 4x2- 6, find m∠ ACB.
8. If AB = 6y and BC = 8y, find BD in terms of y.
B C
DA
E
6 -4
Example 1 Example 2
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Chapter 6 24 Glencoe Geometry
Study Guide and Intervention (continued)
Rectangles
Prove that Parallelograms Are Rectangles The diagonals of a rectangle are congruent, and the converse is also true.
If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
In the coordinate plane you can use the Distance Formula, the Slope Formula, and properties of diagonals to show that a figure is a rectangle.
Quadrilateral ABCD has vertices A(-3, 0), B(-2, 3), C(4, 1), and D(3, -2). Determine whether ABCD is a rectangle.
Method 1: Use the Slope Formula.
slope of −−
AB = 3 - 0 −
-2 - (-3) =
3 −
1 or 3 slope of
−−− AD = -2 - 0
− 3 - (-3)
= -2 −
6 or - 1 −
3
slope of −−−
CD = -2 - 1 −
3 - 4 =
-3 −
-1 or 3 slope of
−−− BC = 1 - 3
− 4 - (-2)
= -2 −
6 or - 1 −
3
Opposite sides are parallel, so the figure is a parallelogram. Consecutive sides are perpendicular, so ABCD is a rectangle.
x
y
O
D
B
A
EC
6 -4
Example
Method 2: Use the Distance Formula.
AB = √ ########## (-3 - (-2) ) 2 + (0 - 3 ) 2 or √ ## 10 BC = √ ######## (-2 - 4 ) 2 + (3 - 1 ) 2 or √ ## 40
CD = √ ######### (4 - 3 ) 2 + (1 - (-2) ) 2 or √ ## 10 AD = √ ########## (-3 - 3 ) 2 + (0 - (-2) ) 2 or √ ## 40
Opposite sides are congruent, thus ABCD is a parallelogram.
AC = √ ######## (-3 - 4 ) 2 + (0 - 1 ) 2 or √ ## 50 BD = √ ########## (-2 - 3 ) 2 + (3 - (-2) ) 2 or √ ## 50
ABCD is a parallelogram with congruent diagonals, so ABCD is a rectangle.
Exercises
COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the
indicated formula.
1. A(−3, 1), B(−3, 3), C(3, 3), D(3, 1); Distance Formula
2. A(−3, 0), B(−2, 3), C(4, 5), D(3, 2); Slope Formula
3. A(−3, 0), B(−2, 2), C(3, 0), D(2 −2); Distance Formula
4. A(−1, 0), B(0, 2), C(4, 0), D(3, −2); Distance Formula
Lesso
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Chapter 6 25 Glencoe Geometry
Skills Practice
Rectangles
ALGEBRA Quadrilateral ABCD is a rectangle.
1. If AC = 2x + 13 and DB = 4x - 1, find DB.
2. If AC = x + 3 and DB = 3x - 19, find AC.
3. If AE = 3x + 3 and EC = 5x - 15, find AC.
4. If DE = 6x - 7 and AE = 4x + 9, find DB.
5. If m∠DAC = 2x + 4 and m∠BAC = 3x + 1, find m∠BAC.
6. If m∠BDC = 7x + 1 and m∠ADB = 9x - 7, find m∠BDC.
7. If m∠ABD = 7x - 31 and m∠CDB = 4x + 5, find m∠ABD.
8. If m∠BAC = x + 3 and m∠CAD = x + 15, find m∠BAC.
9. PROOF: Write a two-column proof.
Given: RSTV is a rectangle and U is the midpoint of
−− VT .
Prove: △RUV $ △SUT
Statements Reasons
COORDINATE GEOMETRY Graph each quadrilateral with the given vertices.
Determine whether the figure is a rectangle. Justify your answer using the
indicated formula.
10. P(-3, -2), Q(-4, 2), R(2, 4), S(3, 0); Slope Formula
11. J(-6, 3), K(0, 6), L(2, 2), M(-4, -1); Distance Formula
12. T(4, 1), U(3, -1), X(-3, 2), Y(-2, 4); Distance Formula
D C
BAE
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Chapter 6 26 Glencoe Geometry
Practice
Rectangles
ALGEBRA Quadrilateral RSTU is a rectangle.
1. If UZ = x + 21 and ZS = 3x - 15, find US.
2. If RZ = 3x + 8 and ZS = 6x - 28, find UZ.
3. If RT = 5x + 8 and RZ = 4x + 1, find ZT.
4. If m∠SUT = 3x + 6 and m∠RUS = 5x - 4, find m∠SUT.
5. If m∠SRT = x + 9 and m∠UTR = 2x - 44, find m∠UTR.
6. If m∠RSU = x + 41 and m∠TUS = 3x + 9, find m∠RSU.
Quadrilateral GHJK is a rectangle. Find each measure if m∠1 = 37.
7. m∠2 8. m∠3
9. m∠4 10. m∠5
11. m∠6 12. m∠7
COORDINATE GEOMETRY Graph each quadrilateral with the given vertices.
Determine whether the figure is a rectangle. Justify your answer using the
indicated formula.
13. B(-4, 3), G(-2, 4), H(1, -2), L(-1, -3); Slope Formula
14. N(-4, 5), O(6, 0), P(3, -6), Q(-7, -1); Distance Formula
15. C(0, 5), D(4, 7), E(5, 4), F(1, 2); Slope Formula
16. LANDSCAPING Huntington Park officials approved a rectangular plot of land for a Japanese Zen garden. Is it sufficient to know that opposite sides of the garden plot are congruent and parallel to determine that the garden plot is rectangular? Explain.
U T
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K
21 5
436
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Chapter 6 27 Glencoe Geometry
1. FRAMES Jalen makes the rectangular
frame shown.
In order to make sure that it is a
rectangle, Jalen measures the distances
BD and AC. How should these two
distances compare if the frame is a
rectangle?
2. BOOKSHELVES A
bookshelf consists of
two vertical planks
with five horizontal
shelves. Are each of the
four sections for books
rectangles? Explain.
3. LANDSCAPING A landscaper is
marking off the corners of a rectangular
plot of land. Three of the corners are in
place as shown.
What are the coordinates of the fourth
corner?
4. SWIMMING POOLS Antonio is
designing a swimming pool on a
coordinate grid. Is it a rectangle?
Explain.
5. PATTERNS Veronica made the pattern
shown out of 7 rectangles with four
equal sides. The side length of each
rectangle is written inside the rectangle.
a. How many rectangles can be formed
using the lines in this figure?
b. If Veronica wanted to extend her
pattern by adding another rectangle
with 4 equal sides to make a larger
rectangle, what are the possible
side lengths of rectangles that she
can add?
Word Problem Practice
Rectangles
1 1
2 3
5
8
5
y
xO
5
5y
xO–5
–5
A B
D C
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Chapter 6 28 Glencoe Geometry
Constant Perimeter
Douglas wants to fence a rectangular region of his back yard for his dog. He bought
200 feet of fence.
1. Complete the table to show the
dimensions of five different rectangular
pens that would use the entire 200 feet
of fence. Then find the area of each
rectangular pen.
2. Do all five of the rectangular pens have
the same area? If not, which one has
the larger area?
3. Write a rule for finding the dimensions of a rectangle with
the largest possible area for a given perimeter.
4. Let x represent the length of a rectangle and y the width.
Write the formula for all rectangles with a perimeter of
200. Then graph this relationship on the coordinate plane
at the right.
Julio read that a dog the size of his new pet, Bennie, should have at least 100 square feet in
his pen. Before going to the store to buy fence, Julio made a table to determine the
dimensions for Bennie’s rectangular pen.
5. Complete the table to find five possible
dimensions of a rectangular fenced area of
100 square feet.
6. Julio wants to save money by purchasing
the least number feet of fencing to enclose the
100 square feet. What will be the dimensions
of the completed pen?
7. Write a rule for finding the dimensions of a rectangle with
the least possible perimeter for a given area.
8. For length x and width y, write a formula for the area of a
rectangle with an area of 100 square feet. Then graph the
formula.
100
100
y
xO
y
xO 100
100
6-4
Perimeter Length Width Area
200 80
200 70
200 60
200 50
200 45
Area Length WidthHow much
fence to buy
100
100
100
100
100
Enrichment
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Chapter 6 29 Glencoe Geometry
Graphing Calculator Activity
TI-Nspire: Exploring Rectangles
A quadrilateral with four right angles is a rectangle. The TI-Nspire can be used to explore some of the characteristics of a rectangle. Use the following steps to draw a rectangle.
Step 1 Set up the calculator in the correct mode.
• Choose Graphs & Geometry from the Home Menu.
• From the View menu, choose 4: Hide Axis
Step 2 Draw the rectangle
• From the 8: Shapes menu choose 3: Rectangle.
• Click once to define the corner of the rectangle. Then move and click again. The side of the rectangle is now defined. Move perpendicularly to draw the rectangle. Click to anchor the shape.
Step 3 Measure the lengths of the sides of the rectangle.
• From the 7: Measurement menu choose 1: Length (Note that when you scroll over the rectangle, the value now shown is the perimeter of the rectangle.)
• Select each endpoint of a segment of the rectangle. Then click or press Enter to anchor the length of the segment in the work area.
• Repeat for the other sides of the rectangle.
Exercises
1. What appears to be true about the opposite sides of the rectangle?
2. Draw the diagonals of the rectangle using 5: Segment from the 6: Points and Lines
Menu. Click on two opposite vertices to draw the diagonal. Repeat to draw the other diagonal.
a. Measure each diagonal using the measurement tool. What do you observe?
b. What is true about the triangles formed by the sides of the rectangle and a diagonal? Justify your conclusion.
3. Press Clear three times and select Yes to clear the screen. Repeat the steps and draw another rectangle. Do the relationships that you found for the first rectangle you drew hold true for this rectangle?
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Chapter 6 30 Glencoe Geometry
Geometer’s Sketchpad Activity
Exploring Rectangles
A quadrilateral with four right angles is a rectangle. The Geometer’s Sketchpad is a useful
tool for exploring some of the characteristics of a rectangle. Use the following steps to draw
a rectangle.
Step 1 Use the Line tool to draw a line anywhere
on the screen.
Step 2 Use the Point tool to draw a point that is
not on the line. To draw a line perpendicular
to the first line you drew, select the first line
and the point. Then choose Perpendicular
Line from the Construct menu.
Step 3 Use the Point tool to draw a point that is
not on either of the lines you have drawn.
Repeat the procedure in Step 2 to draw
lines perpendicular to the two lines you
have drawn.
A rectangle is formed by the segments whose endpoints are the points of intersection of the
lines.
Exercises
Use the measuring capabilities of The Geometer’s Sketchpad to explore
the characteristics of a rectangle.
1. What appears to be true about the opposite sides of the rectangle that you drew? Make a
conjecture and then measure each side to check your conjecture.
2. Draw the diagonals of the rectangle by using the Selection Arrow tool to choose two
opposite vertices. Then choose Segment from the Construct menu to draw the
diagonal. Repeat to draw the other diagonal.
a. Measure each diagonal. What do you observe?
b. What is true about the triangles formed by the sides of the rectangle and a
diagonal? Justify your conclusion.
3. Choose New Sketch from the File menu and follow steps 1–3 to draw another
rectangle. Do the relationships you found for the first rectangle you drew hold true for
this rectangle also?
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Chapter 6 31 Glencoe Geometry
Study Guide and Intervention
Rhombi and Squares
Properties of Rhombi and Squares A rhombus is a quadrilateral with four congruent sides. Opposite sides are congruent, so a rhombus is also a parallelogram and has all of the properties of a parallelogram. Rhombi also have the following properties.
A square is a parallelogram with four congruent sides and four
congruent angles. A square is both a rectangle and a rhombus;
therefore, all properties of parallelograms, rectangles, and rhombi
apply to squares.
In rhombus ABCD, m∠BAC = 32. Find the
measure of each numbered angle.
ABCD is a rhombus, so the diagonals are perpendicular and △ABE
is a right triangle. Thus m∠4 = 90 and m∠1 = 90 - 32 or 58. The
diagonals in a rhombus bisect the vertex angles, so m∠1 = m∠2.
Thus, m∠2 = 58.
A rhombus is a parallelogram, so the opposite sides are parallel. ∠BAC and ∠3 are
alternate interior angles for parallel lines, so m∠3 = 32.
Exercises
Quadrilateral ABCD is a rhombus. Find each value or measure.
1. If m∠ABD = 60, find m∠BDC. 2. If AE = 8, find AC.
3. If AB = 26 and BD = 20, find AE. 4. Find m∠CEB.
5. If m∠CBD = 58, find m∠ACB. 6. If AE = 3x - 1 and AC = 16, find x.
7. If m∠CDB = 6y and m∠ACB = 2y + 10, find y.
8. If AD = 2x + 4 and CD = 4x - 4, find x.
M O
HR
B
CA
D
E
B
32°
1 2
3
4
C
A D
EB
6 -5
The diagonals are perpendicular. −−−
MH ⊥ −−−
RO
Each diagonal bisects a pair of opposite angles. −−−
MH bisects ∠RMO and ∠RHO.
−−−
RO bisects ∠MRH and ∠MOH.
Example
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Chapter 6 32 Glencoe Geometry
Study Guide and Intervention (continued)
Rhombi and Squares
Conditions for Rhombi and Squares The theorems below can help you prove that a parallelogram is a rectangle, rhombus, or square.
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.
If one pair of consecutive sides of a parallelogram are congruent, the parallelogram is a rhombus.
If a quadrilateral is both a rectangle and a rhombus, then it is a square.
Determine whether parallelogram ABCD with
vertices A(−3, −3), B(1, 1), C(5, −3), D(1, −7) is a rhombus, a rectangle, or a square.
AC = √ """""""""" (-3 - 5)2 + ((-3 - (-3))2 = √ "" 64 = 8
BD = √ """""""" (1-1)2 + (-7 - 1)2 = √ "" 64 = 8
The diagonals are the same length; the figure is a rectangle.
Slope of −−
AC = -3 - (-3)
− -3 - 5 =
0 −
-8 = 8 The line is horizontal.
Slope of −−−
BD = 1 - (-7)
− 1 - 1
= 8 −
0 = undefined The line is vertical.
Since a horizontal and vertical line are perpendicular, the diagonals are perpendicular. Parallelogram ABCD is a square which is also a rhombus and a rectangle.
Exercises
Given each set of vertices, determine whether !ABCD is a rhombus, rectangle, or square. List all that apply. Explain.
1. A(0, 2), B(2, 4), C(4, 2), D(2, 0) 2. A(-2, 1), B(-1, 3), C(3, 1), D(2, -1)
3. A(-2, -1), B(0, 2), C(2, -1), D(0, -4) 4. A(-3, 0), B(-1, 3), C(5, -1), D(3, -4)
5. PROOF Write a two-column proof.
Given: Parallelogram RSTU. −−
RS $ −−
ST
Prove: RSTU is a rhombus.
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Chapter 6 33 Glencoe Geometry
Skills Practice
Rhombi and Squares
ALGEBRA Quadrilateral DKLM is a rhombus.
1. If DK = 8, find KL.
2. If m∠DML = 82 find m∠DKM.
3. If m∠KAL = 2x – 8, find x.
4. If DA = 4x and AL = 5x – 3, find DL.
5. If DA = 4x and AL = 5x – 3, find AD.
6. If DM = 5y + 2 and DK = 3y + 6, find KL.
7. PROOF Write a two-column proof.
Given: RSTU is a parallelogram. −−−RX "
−−TX "
−−SX "
−−−UX
Prove: RSTU is a rectangle.
Statements Reasons
COORDINATE GEOMETRY Given each set of vertices, determine whether QRST
is a rhombus, a rectangle, or a square. List all that apply. Explain.
8. Q(3, 5), R(3, 1), S(-1, 1), T(-1, 5)
9. Q(-5, 12), R(5, 12), S(-1, 4), T(-11, 4)
10. Q(-6, -1), R(4, -6), S(2, 5), T(-8, 10)
11. Q(2, -4), R(-6, -8), S(-10, 2), T(-2, 6)
M L
AD K
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Chapter 6 34 Glencoe Geometry
Practice
Rhombi and Squares
PRYZ is a rhombus. If RK = 5, RY = 13 and m∠YRZ = 67, find each measure.
1. KY
2. PK
3. m∠YKZ
4. m∠PZR
MNPQ is a rhombus. If PQ = 3 √ # 2 and AP = 3, find each measure.
5. AQ
6. m∠APQ
7. m∠MNP
8. PM
COORDINATE GEOMETRY Given each set of vertices, determine whether $BEFG
is a rhombus, a rectangle, or a square. List all that apply. Explain.
9. B(-9, 1), E(2, 3), F(12, -2), G(1, -4)
10. B(1, 3), E(7, -3), F(1, -9), G(-5, -3)
11. B(-4, -5), E(1, -5), F(-2, -1), G(-7, -1)
12. TESSELLATIONS The figure is an example of a tessellation. Use a ruler or protractor to measure the shapes and then name the quadrilaterals used to form the figure.
K
P
YR
Z
A
N
M Q
P
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Chapter 6 35 Glencoe Geometry
1. TRAY RACKS A tray rack looks like a
parallelogram from the side. The levels
for the trays are evenly spaced.
What two labeled points form a rhombus
with base AA' ?
2. SLICING Charles cuts a rhombus along
both diagonals. He ends up with four
congruent triangles. Classify these
triangles as acute, obtuse, or right.
3. WINDOWS The edges of a window are
drawn in the coordinate plane.
Determine whether the window is a
square or a rhombus.
4. SQUARES Mackenzie cut a square
along its diagonals to get four congruent
right triangles. She then joined two of
them along their long sides. Show that
the resulting shape is a square.
5. DESIGN Tatianna made the design
shown. She used 32 congruent rhombi to
create the flower-like design at each
corner.
a. What are the angles of the corner
rhombi?
b. What kinds of quadrilaterals are the
dotted and checkered figures?
Word Problem Practice
Rhombi and Squares
y
xO
A
B
C
D
E
F
G
H
B,
C,
D,
E,F,G,
H,
A20 inches
AB = 4 inches,
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Chapter 6 36 Glencoe Geometry
Enrichment
Creating Pythagorean Puzzles
By drawing two squares and cutting them in a certain way, you
can make a puzzle that demonstrates the Pythagorean Theorem.
A sample puzzle is shown. You can create your own puzzle by
following the instructions below.
1. Carefully construct a square and label the length
of a side as a. Then construct a smaller square to
the right of it and label the length of a side as b,
as shown in the figure above. The bases should
be adjacent and collinear.
2. Mark a point X that is b units from the left edge
of the larger square. Then draw the segments
from the upper left corner of the larger square to
point X, and from point X to the upper right corner
of the smaller square.
3. Cut out and rearrange your five pieces to form a
larger square. Draw a diagram to show your
answer.
4. Verify that the length of each side is equal to
√ """ a2 + b2 .
a
a
b X
b
b
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Chapter 6 37 Glencoe Geometry
Study Guide and Intervention
Trapezoids and Kites
Properties of Trapezoids A trapezoid is a quadrilateral with exactly one pair of parallel sides. The midsegment or median of a trapezoid is the segment that connects the midpoints of the legs of the trapezoid. Its measure is equal to one-half the sum of the lengths of the bases. If the legs are congruent, the trapezoid is an isosceles trapezoid. In an isosceles trapezoid both pairs of base angles are congruent and the diagonals are congruent.
The vertices of ABCD are A(-3, -1), B(-1, 3), C(2, 3), and D(4, -1). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid.
slope of −−
AB = 3 - (-1)
− -1 - (-3)
= 4 − 2 = 2
slope of −−−
AD = -1 - (-1)
− 4 - (-3)
= 0 −
7 = 0
slope of −−−
BC = 3 - 3 −
2 - (-1) =
0 −
3 = 0
slope of −−−
CD = -1 - 3 −
4 - 2 = -4
− 2 = -2
Exactly two sides are parallel, −−−
AD and −−−
BC , so ABCD is a trapezoid. AB = CD, so ABCD is an isosceles trapezoid.
Exercises
Find each measure.
1. m∠D 2. m∠L
125°
5
5
40°
COORDINATE GEOMETRY For each quadrilateral with the given vertices, verify that the quadrilateral is a trapezoid and determine whether the figure is an isosceles trapezoid.
3. A(−1, 1), B(3, 2), C(1,−2), D(−2, −1) 4. J(1, 3), K(3, 1), L(3, −2), M(−2, 3)
For trapezoid HJKL, M and N are the midpoints of the legs.
5. If HJ = 32 and LK = 60, find MN.
6. If HJ = 18 and MN = 28, find LK.
T
R U
base
base
Sleg
x
y
O
B
DA
C
6-6
STUR is an isosceles trapezoid.
−−
SR $ −−
TU ; ∠R $ ∠U, ∠S $ ∠T
Example
AB = √ &&&&&&&&&& (-3 - (-1))2 + (-1 - 3)2
= √ &&& 4 + 16 = √ && 20 = 2 √ & 5
CD = √ &&&&&&&&& (2 - 4)2 + (3 - (-1))2
= √ &&& 4 + 16 = √ && 20 = 2 √ & 5
L K
NM
H J
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Chapter 6 38 Glencoe Geometry
Properties of Kites A kite is a quadrilateral with exactly two pairs of consecutive congruent sides. Unlike a parallelogram, the opposite sides of a kite are not congruent or parallel.
The diagonals of a kite are perpendicular.
For kite RMNP, −−−
MP ⊥ −−−
RN
In a kite, exactly one pair of opposite angles is congruent.
For kite RMNP, ∠M $ ∠P
If WXYZ is a kite, find m∠Z.
The measures of ∠Y and ∠W are not congruent, so ∠X $ ∠Z.
m∠X + m∠Y + m∠Z + m∠W = 360
m∠X + 60 + m∠Z + 80 = 360
m∠X + m∠Z = 220
m∠X = 110, m∠Z = 110
If ABCD is a kite, find BC.
The diagonals of a kite are perpendicular. Use the
Pythagorean Theorem to find the missing length.
BP2 + PC2 = BC2
52 + 122 = BC2
169 = BC2
13 = BC
Exercises
If GHJK is a kite, find each measure.
1. Find m∠JRK.
2. If RJ = 3 and RK = 10, find JK.
3. If m∠GHJ = 90 and m∠GKJ = 110, find m∠HGK.
4. If HJ = 7, find HG.
5. If HG = 7 and GR = 5, find HR.
6. If m∠GHJ = 52 and m∠GKJ = 95, find m∠HGK.
Study Guide and Intervention (continued)
Trapezoids and Kites
6-6
Example 1
Example 2
80°
60°
5
5
4
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Chapter 6 39 Glencoe Geometry
ALGEBRA Find each measure.
1. m∠S 2. m∠M
63°
14 14
142°
21 21
3. m∠D 4. RH
36° 70°20
12
12
ALGEBRA For trapezoid HJKL, T and S are midpoints of the legs.
5. If HJ = 14 and LK = 42, find TS.
6. If LK = 19 and TS = 15, find HJ.
7. If HJ = 7 and TS = 10, find LK.
8. If KL = 17 and JH = 9, find ST.
COORDINATE GEOMETRY EFGH is a quadrilateral with vertices E(1, 3), F(5, 0), G(8, −5), H(−4, 4).
9. Verify that EFGH is a trapezoid.
10. Determine whether EFGH is an isosceles trapezoid. Explain.
6-6 Skills Practice
Trapezoids and Kites
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Chapter 6 40 Glencoe Geometry
Practice
Trapezoids and Kites
Find each measure.
1. m∠T 2. m∠Y
60°
68°
7
7
3. m∠Q 4. BC
110° 48°
11 4
7
7
ALGEBRA For trapezoid FEDC, V and Y are midpoints of the legs.
5. If FE = 18 and VY = 28, find CD.
6. If m∠F = 140 and m∠E = 125, find m∠D.
COORDINATE GEOMETRY RSTU is a quadrilateral with vertices R(−3, −3), S(5, 1), T(10, −2), U(−4, −9).
7. Verify that RSTU is a trapezoid.
8. Determine whether RSTU is an isosceles trapezoid. Explain.
9. CONSTRUCTION A set of stairs leading to the entrance of a building is designed in the
shape of an isosceles trapezoid with the longer base at the bottom of the stairs and the
shorter base at the top. If the bottom of the stairs is 21 feet wide and the top is 14 feet
wide, find the width of the stairs halfway to the top.
10. DESK TOPS A carpenter needs to replace several trapezoid-shaped desktops in a
classroom. The carpenter knows the lengths of both bases of the desktop. What other
measurements, if any, does the carpenter need?
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Chapter 6 41 Glencoe Geometry
Word Problem Practice
Trapezoids and Kites
1. PERSPECTIVE Artists use different
techniques to make things appear to be
3-dimensional when drawing in two
dimensions. Kevin drew the walls of a
room. In real life, all of the walls are
rectangles. In what shape did he draw
the side walls to make them appear
3-dimensional?
2. PLAZA In order to give the feeling of
spaciousness, an architect decides to
make a plaza in the shape of a kite.
Three of the four corners of the plaza
are shown on the coordinate plane. If
the fourth corner is in the first
quadrant, what are its coordinates?
y
x
3. AIRPORTS A simplified drawing of
the reef runway complex at Honolulu
International Airport is shown below.
How many trapezoids are there in this
image?
4. LIGHTING A light outside a room shines
through the door and illuminates a
trapezoidal region ABCD on the floor.
A B
D C
Under what circumstances would
trapezoid ABCD be isosceles?
5. RISERS A riser is designed to elevate
a speaker. The riser consists of
4 trapezoidal sections that can be
stacked one on top of the other to
produce trapezoids of varying heights.
20 feet
10 feet
All of the stages have the same height.
If all four stages are used, the width of
the top of the riser is 10 feet.
a. If only the bottom two stages are
used, what is the width of the top
of the resulting riser?
b. What would be the width of the riser
if the bottom three stages are used?
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Chapter 6 42 Glencoe Geometry
Enrichment
Quadrilaterals in Construction
Quadrilaterals are often used in construction work.
1. The diagram at the right represents a roof frame and shows many quadrilaterals. Find the following shapes in the diagram and shade in their edges.
a. isosceles triangle
b. scalene triangle
c. rectangle
d. rhombus
e. trapezoid (not isosceles)
f. isosceles trapezoid
2. The figure at the right represents a window. The wooden part between the panes of glass is 3 inches wide. The frame around the outer edge is 9 inches wide. The outside measurements of the frame are 60 inches by 81 inches. The height of the top and bottom panes is the same. The top three panes are the same size.
a. How wide is the bottom pane of glass?
b. How wide is each top pane of glass?
c. How high is each pane of glass?
3. Each edge of this box has been reinforced with a piece of tape. The box is 10 inches high, 20 inches wide, and 12 inches deep. What is the length of the tape that has been used?
jack rafters
hip rafter
common rafters
Roof Frame
ridge boardvalley rafter
plate
60 in.
81 in.
9 in.
3 in.
3 in.
9 in.
9 in.
10 in.
12 in.
20 in.
6-6
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Assessment
Chapter 6 43 Glencoe Geometry
SCORE 6
Read each question. Then fill in the correct answer.
1. A B C D
2. F G H J
3. A B C D
4. F G H J
5. A B C D
6. F G H J
7. A B C D
Multiple Choice
Student Recording Sheet
Use this recording sheet with pages 452–453 of the Student Edition.
Record your answer in the blank.
For gridded response questions, also enter your answer in the grid by writing
each number or symbol in a box. Then fill in the corresponding circle for that
number or symbol.
8. (grid in)
9.
10.
11.
12. (grid in)
13. 9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
Extended Response
Short Response/Gridded Response
Record your answers for Question 14 on the back of this paper.
8. 12.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
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Chapter 6 44 Glencoe Geometry
6
General Scoring Guidelines
• If a student gives only a correct numerical answer to a problem but does not show how
he or she arrived at the answer, the student will be awarded only 1 credit. All extended-
response questions require the student to show work.
• A fully correct answer for a multiple-part question requires correct responses for all
parts of the question. For example, if a question has three parts, the correct response to
one or two parts of the question that required work to be shown is not considered a fully
correct response.
• Students who use trial and error to solve a problem must show their method. Merely
showing that the answer checks or is correct is not considered a complete response for
full credit.
Exercise 14 Rubric
Score Speci! c Criteria
4
Students correctly determine the quadrilateral in part a is a parallelogram since
opposite sides are congruent. Students correctly determine the quadrilateral in part b
does not contain sufficient information to prove it is a parallelogram. The two horizontal
sides must also be congruent. Students correctly determine that the quadrilateral in
part c is a parallelogram since both pairs of opposite angles are congruent.
3 A generally correct solution, but may contain minor flaws in reasoning or computation.
2 A partially correct interpretation and/or solution to the problem.
1 A correct solution with no evidence or explanation.
0An incorrect solution indicating no mathematical understanding of the concept or task,
or no solution is given.
Rubric for Scoring Extended Response
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Assessm
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NAME DATE PERIOD
SCORE
1. Find the sum of the measures of the interior angles of a convex 70-gon.
2. The measure of each interior angle of a regular polygon is 172. Find the number of sides in the polygon.
3. The measure of each exterior angle of a regular polygon is 18. Find the number of sides in the polygon.
4. Given parallelogram ABCD with C(5, 4), find the coordinates of A if the diagonals
−− AC and
−−− BD intersect at (2, 7).
5. MULTIPLE CHOICE Find m∠1 in parallelogram ABCD.
A 64 C 46
B 58 D 36
1.
2.
3.
4.
5.
1. Determine whether this quadrilateral is a parallelogram. Justify your answer.
For Questions 2–4, write true or false.
2. A quadrilateral with two pairs of parallel sides is always a parallelogram.
3. The diagonals of a parallelogram are always perpendicular.
4. The slope of −−
AB and −−−
CD is 3 −
5 and the slope of
−−− BC and
−−− AD is
- 5 −
3 . ABCD is a parallelogram.
5. Refer to parallelogram ABCD. If AB = 8 cm, what is the perimeter of the parallelogram?
1.
2.
3.
4.
5.
Chapter 6 Quiz 1(Lessons 6-1 and 6-2)
Chapter 6 Quiz 2(Lesson 6-3)
6
6
A
D
B
C
6 cm
Chapter 6 45 Glencoe Geometry
110°
46°1
D C
BA
SCORE
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NAME DATE PERIOD
Chapter 6 46 Glencoe Geometry
SCORE
1. MULTIPLE CHOICE RSTV is a rhombus. Which of the following statements is NOT true?
A −−
RV " −−
TS
B −−
RV ⊥ −−
TS
C −−
RS ‖ −−
TV
D ∠R " ∠T
For Questions 2 and 3, refer to trapezoid MNPQ.
2. Find m∠M.
3. Find m∠Q.
4. True or false. A quadrilateral that is a rectangle and a rhombus is a square.
5. &ABCD has vertices A(4, 0), B(0, 4), C(−4, 0), and D(0, −4). Determine whether ABCD is a rectangle, rhombus, or square. List all that apply.
1.
2.
3.
4.
5.
For Questions 1 and 2, refer to kite DEFC.
1. If m∠DCF = 34 and m∠DEF = 90,
find m∠CDE.
2. If DR = 5 and RE = 5, find FE.
For Questions 3 and 4, refer to trapezoid NPQM
where X and Y are midpoints of the sides.
3. If MQ = 15 and XY = 10, find NP.
4. If NP = 13 and MQ = 18, find XY.
5. If CDEF is a trapezoid with vertices C(0, 2), D(2, 4), E(7, 3), and F(1, −3), how can you prove that it is an isosceles trapezoid?
1.
2.
3.
4.
5.
6 Chapter 6 Quiz 3(Lessons 6-4 and 6-5)
Chapter 6 Quiz 4(Lesson 6-6)
(2x + 3)°
(x + 9)°
6 SCORE
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Assessment
Chapter 6 47 Glencoe Geometry
6 SCORE
1. Find the measure of each exterior angle of a regular 56-gon. Round to the nearest tenth.
A 3.2 B 6.4 C 173.6 D 9720
2. Given BE = 2x + 6 and ED = 5x - 12 in parallelogram ABCD, find BD.
F 6 H 18
G 12 J 36
3. If the slope of −−−
PQ is 2 − 3 and the slope of
−−− QR is - 1 −
2 , find the slope of
−− SR so that
PQRS is a parallelogram.
A 2 −
3 B
3 −
2 C -
1 −
2 D 2
4. Find m∠W in parallelogram RSTW.
F 17 H 55
G 33 J 125
5. Find the sum of the measures of the interior angles of a convex 48-gon.
A 172.5 B 360 C 8280 D 8640
D
ECB
A
6.
7.
8.
9.
10.
1.
2.
3.
4.
5.
Part II
6. Find x.
7. ABCD is a parallelogram with m∠A = 138. Find m∠B.
8. Determine whether ABCD is a parallelogram if AB = 6, BC = 12, CD = 6, and DA = 12. Justify your answer.
9. In parallelogram MLKJ, find m∠MLK and m∠LKJ.
10. XYWZ is a quadrilateral with vertices W(1, −4), X(-4, 2), Y(1, −1), and Z(−2, −3). Determine if the quadrilateral is a parallelogram. Use slope to justify your answer.
Part I Write the letter for the correct answer in the blank at the right of each question.
Chapter 6 Mid-Chapter Test(Lessons 6-1 through 6-3)
(3x + 2)°(2x - 12)°
(x - 10)°(x + 30)°
(2x - 1)°
(x + 16)°
18°
12°
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Chapter 6 48 Glencoe Geometry
6 SCORE
Choose from the terms above to complete each sentence.
1. A quadrilateral with only one pair of opposite sides parallel and the other pair of opposite sides congruent is a(n) .
2. A quadrilateral with two pairs of opposite sides parallel is a(n)
.
3. A quadrilateral with only one pair of opposite sides parallel is a(n) .
4. A quadrilateral that is both a rectangle and a rhombus is a(n) .
5. A quadrilateral with four congruent sides is a(n) .
Write whether each sentence is true or false. If false,
replace the underlined word or number to make a true
sentence.
6. A quadrilateral with four right angles is a .
7. A quadrilateral with two pairs of congruent consecutive sides is a .
Choose the correct term to complete each sentence.
8. Segments that join opposite vertices in a quadrilateral are called (medians, diagonals).
9. The segment joining the midpoints of the nonparallel sides of a trapezoid is called the (median, diagonal).
Define each term in your own words.
10. base angles of an isosceles trapezoid
11. legs of a trapezoid
? 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
base
base angle
diagonal
isosceles trapezoid
legs
midsegment of a trapezoid
parallelogram
rectangle
rhombus
square
trapezoid
Chapter 6 Vocabulary Test
?
?
?
?
trapezoid
kite
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Assessment
Chapter 6 49 Glencoe Geometry
6 SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. Find the sum of the measures of the interior angles of a convex 30-gon.
A 5400 B 5040 C 360 D 168
2. Find the sum of the measures of the exterior angles of a convex 21-gon.
F 21 G 180 H 360 J 3420
3. If the measure of each interior angle of a regular polygon is 108, find the measure of each exterior angle.
A 18 B 72 C 90 D 108
4. For parallelogram ABCD, find the value of x.
F 4 H 16
G 10.25 J 21.5
5. Which of the following is a property of a parallelogram?
A The diagonals are congruent. C The diagonals are perpendicular.
B The diagonals bisect the angles. D The diagonals bisect each other.
6. Find the values of x and y so that ABCD will be a parallelogram.
F x = 6, y = 42
G x = 6, y = 22
H x = 20, y = 42
J x = 20, y = 22
7. Find the value of x so that this quadrilateral is a parallelogram.
A 44 C 90
B 46 D 134
8. Parallelogram ABCD has vertices A(0, 0), B(2, 4), and C(10, 4). Find the coordinates of D.
F D(8, 0) G D(10, 0) H D(0, 4) J D(10, 8)
9. Which of the following is a property of all rectangles?
A four congruent sides C diagonals are perpendicular
B diagonals bisect the angles D four right angles
10. ABCD is a rectangle with diagonals −−
AC and −−−
BD . If AC = 2x + 10 and
BD = 56, find the value of x.
F 23 G 33 H 78 J 122
11. ABCD is a rectangle with B(-5, 0), C(7, 0) and D(7, 3). Find the coordinates of A.
A A(-5, 7) B A(3, 5) C A(-5, 3) D A(7, -3)
x° 134°
134° 46°
(4x)°
(y - 10)°32°
24°
5x - 12
3x + 20
A D
CB
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Chapter 6 Test, Form 1
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Chapter 6 50 Glencoe Geometry
6
12. For rhombus ABCD, find m∠1.
F 45 H 90
G 60 J 120
13. Find m∠PRS in square PQRS.
A 30 C 60
B 45 D 90
14. Choose a pair of base angles of trapezoid ABCD.
F ∠A, ∠C H ∠A, ∠D
G ∠B, ∠D J ∠D, ∠C
15. In trapezoid DEFG, find m∠D.
A 44 C 108
B 72 D 136
16. The hood of Olivia’s car is the shape of a trapezoid. The base bordering the
windshield measures 30 inches and the base at the front of the car measures
24 inches. What is the width of the median of the hood?
F 25 in. G 27 in. H 28 in. J 29 in.
17. The length of one base of a trapezoid is 44, the median is 36, and the other
base is 2x + 10. Find the value of x.
A 9 B 17 C 21 D 40
18. Given trapezoid ABCD with median −−
EF , which
of the following is true?
F EF = 1 − 2 AD H AB = EF
G AE = FD J EF = BC + AD −
2
19. PQRS is a kite. Find m∠S. A 100 C 200
B 160 D 360
20. JKLM is a kite, find JM.
F √ $$ 29 H √ $$ 13
G √ $$ 89 J 11
Bonus Find x and m∠WYZ in
rhombus XYZW.
X Y
ZW
(3x + 20)°
(x 2)°
DA
B C
FE
D G
E F136°
72°
P S
Q R
1
A D
CB
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
Chapter 6 Test, Form 1 (continued)
D C
A B
124° 36°
8 2
5
5
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Assessment
Chapter 6 51 Glencoe Geometry
6 SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. Find the sum of the measures of the interior angles of a convex 45-gon.
A 8100 B 7740 C 360 D 172
2. Find the value of x.
F 30 H 102
G 66 J 138
3. Find the sum of the measures of the exterior angles of a convex 39-gon.
A 39 B 90 C 180 D 360
4. Which of the following is a property of a parallelogram?
F Each pair of opposite sides is congruent.
G Only one pair of opposite angles is congruent.
H Each pair of opposite angles is supplementary.
J There are four right angles.
5. For parallelogram ABCD, find m∠1.
A 60 C 36
B 54 D 18
6. ABCD is a parallelogram with diagonals intersecting at E. If AE = 3x + 12 and EC = 27, find the value of x.
F 5 G 17 H 27 J 47
7. Find the values of x and y so that this quadrilateral is a parallelogram.
A x = 13, y = 24 C x = 7, y = 24
B x = 13, y = 6 D x = 7, y = 6
8. Find the value of x so that this quadrilateral is a parallelogram.
F 12 H 36
G 24 J 132
9. Parallelogram ABCD has vertices A(8, 2), B(6, -4), and C(-5, -4). Find the coordinates of D.
A D(-5, 2) B D(-3, 2) C D(-2, 2) D D(-4, 8)
10. ABCD is a rectangle. If AC = 5x + 2 and BD = x + 22, find the value of x.
F 5 G 6 H 11 J 26
11. Which of the following is true for all rectangles?
A The diagonals are perpendicular.
B The diagonals bisect the angles.
C The consecutive sides are congruent.
D The consecutive sides are perpendicular.
(2x + 60)°
(4x - 12)°
(2x + 30)°
(5x - 9)°
(y - 12)°(1–2y)°
120°
36°
1
A D
CB
(2x + 10)°
(x + 40)°
(x - 20)°x°
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Chapter 6 Test, Form 2A
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Chapter 6 52 Glencoe Geometry
6
12. ABCD is a rectangle with B(-4, 6), C(-4, 2), and D(10, 2). Find the coordinates of A.
F A(6, 4) G A(10, 4) H A(2, 6) J A(10, 6)
13. For rhombus GHJK, find m∠1.
A 22 C 68
B 44 D 90
14. The diagonals of square ABCD intersect at E. If AE = 2x + 6 and BD = 6x - 10, find AC.
F 11 G 28 H 56 J 90
15. ABCD is an isosceles trapezoid with A(10, -1), B(8, 3), and C(-1, 3). Find the coordinates of D.
A D(-3, -1) B D(-10, -11) C D(-1, 8) D D(-3, 3)
16. For isosceles trapezoid MNOP, find m∠MNP.
F 44 H 80
G 64 J 116
17. The length of one base of a trapezoid is 19 inches and the length of the median is 16 inches. Find the length of the other base.
A 35 in. B 19 in. C 17.5 in. D 13 in.
18. Judith built a fence to surround her property. On a coordinate plane, the four
corners of the fence are located at (-16, 1), (-6, 5), (4, 1), and (-6, -3). Which of the following most accurately describes the shape of Judith’s fence?
F square H rhombus
G rectangle J trapezoid
19. For kite PQRS, find m∠S
A 248 C 112
B 68 D 124
20. ABCD is a parallelogram with coordinates A(4, 2), B(4, -1), C(-2, -1), and D(-2, 2). To prove that ABCD is a rectangle, you would plot the parallelogram on a coordinate plane and then find which of the following?
F measures of the angles H slopes of the diagonals
G lengths of the diagonals J midpoints of the diagonals
Bonus Find the possible value(s) of x in rectangle JKLM.
J K
LM
x 2 + 8
3x + 36
P
ON
M36°
44°
22°
1G H
JK
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
Chapter 6 Test, Form 2A (continued)
22°
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Assessment
Chapter 6 53 Glencoe Geometry
6 SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. Find the sum of the measures of the interior angles of a convex 50-gon.
A 9000 B 8640 C 360 D 172.8
2. Find the value of x.
F 16 H 50
G 34 J 70
3. Find the sum of the measures of the exterior angles of a convex 65-gon.
A 5.54 B 90 C 180 D 360
4. Which of the following is a property of all parallelograms?
F Each pair of opposite angles is congruent.
G Only one pair of opposite sides is congruent.
H Each pair of opposite angles is supplementary.
J There are four right angles.
5. For parallelogram ABCD, find m∠1.
A 19 C 52
B 38 D 56
6. ABCD is a parallelogram with diagonals intersecting at E. If AE = 4x - 8 and EC = 36, find the value of x.
F 7 G 11 H 15.5 J 38
7. Find the values of the values of x and y so that the quadrilateral is a parallelogram.
A x = 27, y = 90 C x = 13, y = 90
B x = 27, y = 40 D x = 13, y = 40
8. Find the value of x so that the quadrilateral is a parallelogram.
F 7 1 − 3 H 12
G 8 J 66
9. ABCD is a parallelogram with A(5, 4), B(-1, -2), and C(8, -2). Find the coordinates of D.
A D(-5, 4) B D(8, 2) C D(14, 4) D D(4, 1)
10. ABCD is a rectangle. If AB = 7x - 6 and CD = 5x + 30, find the value of x.
F 5 1 − 3 G 12 H 13 J 18
11. Which of the following is true for all rectangles?
A The diagonals are perpendicular.
B The consecutive angles are supplementary.
C The opposite sides are supplementary.
D The opposite angles are complementary.
6x - 6 3x + 30
(4x - 4)°
(3x + 23)°
(y - 60)°
1–3y °
38°
124°1
A D
B C
(2x - 10)°
(x + 30)°
(x + 70)° x°
(2x)°
(2x)°
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Chapter 6 Test, Form 2B
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Chapter 6 54 Glencoe Geometry
6
12. ABCD is a rectangle with B(-7, 3), C(5, 3), and D(5, -8). Find the
coordinates of A.
F A(-8, -7) G A(-7, -8) H A(-5, -3) J A(-8, -5)
13. For rhombus GHJK, find m∠1.
A 90 C 52
B 64 D 38
14. The diagonals of square ABCD intersect at E. If AE = 3x - 4 and BD = 10x - 48, find AC.
F 90 G 52 H 26 J 10
15. ABCD is an isosceles trapezoid with A(0, -1), B(-2, 3), and D(6, -1). Find the coordinates of C.
A C(6, 1) B C(9, 4) C C(2, 3) D C(8, 3)
16. For isosceles trapezoid MNOP, find m∠MNP.
F 42 H 82
G 70 J 98
17. The length of one base of a trapezoid is 19 meters and the length of the median is 23 meters. Find the length of the other base.
A 15 m B 21 m C 27 m D 42 m
18 On a coordinate plane, the four corners of Ronald’s garden are located at (0, 2), (4, 6), (8, 2), and (4, -2). Which of the following most accurately describes the shape of Ronald’s garden?
F square H rhombus
G rectangle J trapezoid
19. For kite WXYZ, find m∠W.
A 106 C 212
B 148 D 360
20. ABCD is a parallelogram with coordinates A(4, 2), B(3, −1), C(−1, −1), andD(−1, 2). To prove that ABCD is a rhombus, you would plot the parallelogram on a coordinate plane and then find which of the following?
F measures of the angles H slopes of the diagonals
G lengths of the diagonals J midpoints of the diagonals
Bonus The sum of the measures of the interior angles of a convex polygon is ten times the sum of the measures of its exterior angles. Find the number of sides of the polygon.
28°
42°
P
ON
M
128°
1
G K
JH
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
130° 18°
Chapter 6 Test, Form 2B (continued)
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NAME DATE PERIOD
Assessment
Chapter 6 55 Glencoe Geometry
6 SCORE
1. What is the sum of the interior angles of an octagonal box?
2. A convex pentagon has interior angles with measures (5x - 12)°, (2x + 100)°, (4x + 16)°, (6x + 15)°, and (3x + 41)°. Find the value of x.
3. If the measure of each interior angle of a regular polygon is 171, find the number of sides in the polygon.
4. In parallelogram ABCD,
m∠1 = x + 12, and m∠2 = 6x - 18.
Find m∠1.
5. Find the measure of each exterior angle of a regular 45-gon.
6. In parallelogram ABCD, m∠A = 58. Find m∠B.
7. Find the coordinates of the intersection of the diagonals of parallelogram XYZW with vertices X(2, 2), Y(3, 6), Z(10, 6), and W(9, 2).
8. Determine whether ABCD is a parallelogram. Justify your answer.
9. Determine whether the quadrilateral with vertices A(5, 7), B(1, -2), C(-6, -3), and D(2, 5) is a parallelogram. Use the slope formula.
10. For quadrilateral ABCD, the slope of −−
AB is 1 − 4 , the slope of
−−− BC
is - 2
− 3
, and the slope of −−−
CD is 1 − 4 . Find the slope of
−−− DA so that
ABCD will be a parallelogram.
11. Given rectangle ABCD, find the value of x.
12. ABCD is a parallelogram and −−
AC # −−−
BD . Determine whether ABCD is a rectangle. Justify your answer.
13. ABCD is a rhombus with diagonals intersecting at E. If
m∠ABC is three times m∠BAD, find m∠EBC.
A B
CD
(2x + 10)°
(3x - 30)°
A D
CB
12 C
D
A
B
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Chapter 6 Test, Form 2C
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Chapter 6 56 Glencoe Geometry
6
14. TUVW is a square with U(10, 2), V(8, 8), and W(2, 6). Find the
coordinates of T.
15. For isosceles trapezoid MNOP, find
m∠MNQ.
16. ABCD is a quadrilateral with vertices A(8, 3), B(6, 7), C(-1, 5), and D(-6, -1). Determine whether ABCD is a trapezoid. Justify your answer.
17. The length of the median of trapezoid EFGH is 13 feet. If the bases have lengths 2x + 4 and 10x - 50, find x.
18. ABCD is a kite, If RC = 10, and BD = 48, find CD.
For Questions 19–25, write true or false.
19. A rectangle is always a parallelogram.
20. The diagonals of a rhombus are always perpendicular.
21. The diagonals of a square always bisect each other.
22. A trapezoid always has two congruent sides.
23. The median of a trapezoid is always parallel to the bases.
24. A kite has exactly two congruent angles.
25. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rectangle.
Bonus In parallelogram ABCD, AB = 2x - 7, BC = x + 3y, CD = x + y, and AD = 2x - y - 1. Find the values of x and y.
N
Q
O
PM59°
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
B:
Chapter 6 Test, Form 2C (continued)
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NAME DATE PERIOD
Assessment
Chapter 6 57 Glencoe Geometry
6 SCORE
1. Bruce is building a tabletop in the shape of an octagon. Find the sum of the external angles of the tabletop.
2. A convex octagon has interior angles with measures (x + 55)°, (3x + 20)°, 4x°, (4x - 10)°, (6x - 55)°, (3x + 52)°, 3x°, and (2x + 30)°. Find the value of x.
3. If the measure of each interior angle of a regular polygon is 176 find the number of sides in the polygon.
4. In parallelogram ABCD,
m∠1 = x + 25, and m∠2 = 2x.
Find m∠2.
5. Find the measure of each exterior angle of a regular 100-gon.
6. In parallelogram ABCD, m∠A = 63. Find m∠B.
7. Find the coordinates of the intersection of the diagonals of parallelogram XYZW with vertices X(3, 0), Y(3, 8), Z(-2, 6), and W(-2, -2).
8. Determine whether this quadrilateral is a parallelogram. Justify your answer.
9. Determine whether a quadrilateral with vertices A(5, 7), B(1, -1), C(-6, -3), and D(-2, 5) is a parallelogram. Use the slope formula.
10. If the slope of −−
AB is 1 − 2 , the slope of
−−− BC is -4, and the slope of
−−− CD is 1 −
2 , find the slope of
−−− DA so that ABCD is a parallelogram.
11. For rectangle ABCD, find the value of x.
12. ABCD is a parallelogram and m∠A = 90. Determine whether ABCD is a rectangle. Justify your answer.
B C
DA
(6x + 20)°
(3x - 2)°
1
2A D
CB
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Chapter 6 Test, Form 2D
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Chapter 6 58 Glencoe Geometry
6
13. ABCD is a rhombus with diagonals intersecting at E. If
m∠ABC is four times m∠BAD, find m∠EBC.
14. PQRS is a square with Q(-2, 8), R(5, 7), and S(4, 0). Find the coordinates of P.
15. For isosceles trapezoid MNOP, find m∠MNQ.
16. ABCD is a quadrilateral with A(8, 21), B(10, 27), C(26, 26), and D(18, 2). Determine whether ABCD is a trapezoid. Justify your answer.
17. The length of the median of trapezoid EFGH is 17 centimeters. If the bases have lengths 2x + 4 and 8x - 50, find the value of x.
18. For kite ABCD, if RA = 15, and BD = 16, find AD.
For Questions 19–25, write true or false.
19. A parallelogram always has four right angles.
20. The diagonals of a rhombus always bisect the angles.
21. A rhombus is always a square.
22. A rectangle is always a square.
23. The diagonals of an isosceles trapezoid are always congruent.
24. The median of a trapezoid always bisects the angles.
25. The diagonals of a kite are always perpendicular.
Bonus The measure of each interior angle of a regular polygon is 24 more than 38 times the measure of each exterior angle. Find the number of sides of the polygon.
M P
ON
Q74°
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
B:
Chapter 6 Test, Form 2D (continued)
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NAME DATE PERIOD
Assessment
Chapter 6 59 Glencoe Geometry
6 SCORE
1. The sum of the interior angles of an animal pen is 900°. How many sides does the pen have?
2. A convex hexagon has interior angles with measures x°, (5x - 103)°, (2x + 60)°, (7x - 31)°, (6x - 6)°, and (9x - 100)°. Find the value of x and the measure of each angle.
3. Find the measure of each exterior angle of a regular 2x-gon.
4. For parallelogram ABCD, find m∠1.
5. ABCD is a parallelogram with diagonals that intersect each other at E. If AE = x2 and EC = 6x - 8, find all possible values of AC.
6. Determine whether the quadrilateral is a parallelogram. Justify your answer.
7. For quadrilateral ABCD, the slope of −−
AB is 2 − 3 and the slope
of −−−
BC is -2. Find the slopes of −−−
CD and −−−
DA so that ABCD will be a parallelogram.
8. In rectangle ABCD, find m∠1.
9. The diagonals of rhombus ABCD intersect at E. If
m∠BAE = 2 − 3 (m∠ABE), find m∠BCD.
10. The diagonals of square ABCD intersect at E. If AE = 2, find the perimeter of ABCD.
11. For isosceles trapezoid ABCD, find AE.
12. Points G and H are midpoints of −−
AF and −−−
DE inregular hexagon ABCDEF. If AB = 6 find GH.
13. The vertices of trapezoid ABCD are A(10, -1), B(6, 6), C(-2, 6), and D(-8, -1). Find the length of the median.
B C
F
A D
HG
E
A D
CB
E F60°
8
B C
DA
(3x - 16)°
1(2x + 10)°
96° 31°1
D C
BA
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Chapter 6 Test, Form 3
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Chapter 6 60 Glencoe Geometry
6
14.
15.
16.
17.
18.
19.
20.
B:
14. Determine whether the quadrilateral ABCD with vertices A(0,−1), B(−4,−3), C(−5, 1), D(1, 7) is a kite. Justify your answer.
15. Determine whether the quadrilateral ABCD with vertices A(6, 2), B(2, 10), C(-6, 6), and D(-2, -2) is a rectangle. Justify your answer.
16. Determine whether quadrilateral ABCD with vertices A(1, 6), B(7, 6), C(2, -3), and D(-4, -3) is a parallelogram. Use the distance formula.
17. Find the value of x in kite EFGH.
For Questions 18 and 19, complete
the two-column proof by supplying
the missing information for each
corresponding location.
Given: ABCD is a parallelogram.
−−−
BQ −−−
DS , −−
PA −−−
RC
Prove: PQRS is a parallelogram.
Statements Reasons
1. ABCD is a #.
2. −−
AD −−
CB
3. −−
PA −−
RC
4. −−
PD −−
RB
5. −−
AB −−
CD
6. −−
BQ −−
DS
7. −−
AQ −−
CS
8. ∠B ∠D, ∠A ∠C
9. △QBR △SDP, △PAQ △RCS
10. −−
QP −−
RS , −−
QR −−
PS
11. PQRS is a parallelogram.
1. Given
2. (Question 18)
3. Given
4. Seg. Sub. Prop.
5. Opp. sides of a # are .
6. Given
7. Seg. Sub. Prop.
8. Opp. & of a # are .
9. SAS
10. CPCTC
11. (Question 19)
20. In isosceles trapezoid ABCD, AE = 2x + 5, EC = 3x - 12, and BD = 4x + 20. Find the value of x.
Bonus If three of the interior angles of a convex hexagon each measure 140, a fourth angle measures 84, and the measure of the fifth angle is 3 times the measure of the sixth angle, find the measure of the sixth angle.
(x + 5)°
x°
(140 - x)°
B C
EDA
A P
R
D
C
SQ
B
Chapter 6 Test, Form 3 (continued)
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NAME DATE PERIOD
Assessment
Chapter 6 61 Glencoe Geometry
6 SCORE Chapter 6 Extended-Response Test
Demonstrate your knowledge by giving a clear, concise solution to
each problem. Be sure to include all relevant drawings and justify
your answers. You may show your solution in more than one way or
investigate beyond the requirements of the problem.
1. a. Draw a regular convex polygon and a convex polygon that is not
regular, each with the same number of sides.
b. Label the measures of each exterior angle on your figures.
c. Find the sum of the exterior angles for each figure. What conjecture can
be made?
2. Draw a rectangle. Connect the midpoints of the consecutive sides. What
type of quadrilateral is formed? How do you know?
3. Draw an example to show why one pair of opposite sides congruent and the
other pair of opposite sides parallel is not sufficient to form a
parallelogram.
4. a. Name a property that is true for a square and not always true for a
rectangle.
b. Name a property that is true for a square and not always true for a
rhombus.
c. Name a property that is true for a rectangle and not always true for a
parallelogram.
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Chapter 6 62 Glencoe Geometry
6 SCORE
1. Find the coordinates of X if V(0.5, 5) is the midpoint of −−− UX with
U(15, 21). (Lesson 1-3)
A (-14, -11) C (0, 0)
B (7.75, 22.5) D (15.5, -5)
2. Which of the following are possible measures for vertical angles G and H? (Lesson 2-8)
F m∠G = 125 and m∠H = 55
G m∠G = 125 and m∠H = 125
H m∠G = 55 and m∠H = 45
J m∠G = 55 and m∠H = 152.5
3. Determine which lines are parallel. (Lesson 3-5)
A # $% NS ‖ # $% PT C # $% QR ‖ # $% ST
B # $% NP ‖ # $% ST D # $% NP ‖ # $$% QR
4. Find the coordinates of B, the midpoint of −− AC , if A(2a, b) and
C(0, 2b). (Lesson 4-8)
F (2a, 2b) G (a, b) H (a, 3 −
2 b) J (
3 −
2 a, b)
5. If −− RV is an angle bisector, find
m∠UVT. (Lesson 5-1)
A 10 C 68
B 34 D 136
6. Find the slope of the line that passes through points A(-7, 14) and B(5, -2). (Lesson 3-3)
F - 4 − 3 G -
3 −
4 H
3 −
4 J
4 −
3
7. Which statement ensures that quadrilateral QRST is a parallelogram? (Lesson 6-3)
A ∠Q ' ∠S C −−− QT ‖
−− RS
B −−− QR '
−− TS and −−− QR ‖
−− TS D m∠Q + m∠S = 180
Q R
ST
(2y + 14)°
(4y - 6)°
T VS
R
U
67°
113°
65°
N P
ST
RQ
1. A B C D
2. F G H J
3. A B C D
4. F G H J
5. A B C D
6. F G H J
7. A B C D
Standardized Test Practice(Chapters 1–6)
Part 1: Multiple Choice
Instructions: Fill in the appropriate circle for the best answer.
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NAME DATE PERIOD
Assessment
Chapter 6 63 Glencoe Geometry
6
8. What is the equation of the line that contains (-12, 9) and is perpendicular to the line y =
2 −
3 x + 5? (Lesson 3-4)
F y = - 3 − 2 x - 9 H y = - 2 −
3 x - 1
G y = 3 − 2 x - 1 J y = 2 −
3 x + 17
9. Which of the following theorems can be used to prove △ABC # △DEC? (Lesson 4-5)
A SSS C SAS
B AAS D ASA
10. What is the value of x? (Lesson 6-6)
F 2 H 5.5
G 4 J 7
11. For △ABC, AB = 6 and BC = 17. Which of the following is a possible length for
−−− AC ? (Lesson 5-3)
A 5 B 9 C 13 D 24
12. What is m∠T in kite STVW?
F 100 H 95
G 130 J 260
A
D
B C
E
8. F G H J
9. A B C D
10. F G H J
11. A B C D
12. F G H J
13. If △UVW is an isosceles triangle, −−−
UV # −−−
WU , UV = 16b - 40, VW = 6b, and WU = 10b + 2, find the value of b. (Lesson 4-1)
14. Find the sum of the measures of the interior angles for a convex heptagon. (Lesson 6-1)
13.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
14.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
Standardized Test Practice (continued)
4x - 4
16
20
Part 2: Gridded Response
Instructions: Enter your answer by writing each digit of the answer
in a column box and then shading in the appropriate circle that
corresponds to that entry.
85° 15°
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Chapter 6 64 Glencoe Geometry
6
15. A polygon has six congruent sides. Lines containing two of its sides contain points in its interior. Name the polygon by its number of sides, and then classify it as convex or concave and regular or irregular. (Lesson 1-6)
16. If −−
RT " −−−
QM and RT = 88.9 centimeters, find QM. (Lesson 2-7)
17. Which segment is the shortest segment
from D to # $$% JM ? (Lesson 5-2)
18. If △ABC " △WXY, AB = 72, BC = 65, CA = 13, XY = 7x - 12, and WX = 19y + 34, find the values of x and y. (Lesson 4-3)
19. Freda bought two bells for just over $90 before tax. State the assumption you would make to write an indirect proof to show that at least one of the bells costs more than $45. (Lesson 5-4)
20. The area of the base of a cylinder is 5 cm2 and the height of the cylinder is 8 cm. Find the volume of the cylinder. (Lesson 1-7)
21. JKLM is a kite. Complete each statement. (Lesson 6-6)
a. −−−
MJ "
b. −−−
MK ⊥
c. m∠L = m∠
D
J K L H M
15.
16.
17.
18.
19.
20.
21. a.
b.
c.
Part 3: Short Response
Instructions: Place your answers in the space provided.
6
6
Standardized Test Practice (continued)
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Assessment
Chapter 6 65 Glencoe Geometry
6 SCORE
1. Use a protractor to classify △UVW, △UWX, and △XWY as acute, equiangular, obtuse, or right.
2. In the figure, ∠1 " ∠2. Find the measures of the numbered angles.
3. Name the corresponding congruent sides for △AFP " △STX.
4. Determine whether △ABC " △PQR given A(2, -7), B(5, 3), C(-4, 6), P(8, -1), Q(11, 9), and R(2, 12).
5. In the figure, −−
LK bisects ∠JKM and ∠KLJ " ∠KLM. Determine which theorem or postulate can be used to prove that △JKL " △MKL.
6. Triangle ABC is isosceles with AB = BC. Name a pair of congruent angles in this triangle.
7. For kite WXYZ, find m∠Z.
For Questions 8 and 9, refer to the figure.
8. Find the value of a and m∠ZWT if −−−
ZW is an altitude of △XYZ, m∠ZWT = 3a + 5, and m∠TWY = 5a + 13.
9. Determine which angle has the greatest measure: ∠YWZ, ∠WZY, or ∠ZYW.
10. Mr. Ramirez bought a stove and a dishwasher for just over $1206. State the assumption you would make to start an indirect proof to show that at least one of the appliances cost more than $603.
XW
Y
T
Z
K L
J
M
65°
70°
110°13
2
D
E F H
G
V
U
W
X
Y
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Unit 2 Test(Chapters 4 – 6)
43°95°
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Chapter 6 66 Glencoe Geometry
6
11. Determine whether 128 feet, 136 feet, and 245 feet can be the lengths of the sides of a triangle.
12. Write an inequality to describe the possible values of x.
13. The measure of an interior angle of a regular polygon is 140. Find the number of sides in the polygon.
14. For parallelogram JKMH, find
m∠JHK, m∠HMK, and the value of x.
15. Determine whether the vertices of quadrilateral DEFG form a parallelogram given D(-3, 5), E(3, 6), F(-1, 0), and G(6, 1).
16. For rectangle WXYZ with diagonals −−−
WY and −−
XZ ,
WY = 3d + 4 and XZ = 4d - 1, find the value of d.
17. If m∠BEC = 9z + 45 in rhombus ABCD, find the value of z.
18. In trapezoid HJLK, M and N are midpoints of the legs. Find KL.
19. Prove that quadrilateral PQRS isNOT a parallelogram.
JH
K L
NM
45
28
A C
B
D
E
J
K
M
H20°
3x + 8
7x - 24
52°
2x - 512
12
14
11
14
41˚38˚
11.
12.
13.
14.
15.
16.
17.
18.
19.
x
y
-1
1
2
3
4
5
1 2 3 4-2-3-4
Unit 2 Test (continued)
Answers
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Chapter 6 A1 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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ge f
rom
th
e f
irst
colu
mn
?
•
F
or
those
sta
tem
en
ts t
hat
you
mark
wit
h a
D,
use
a p
iece
of
pap
er
to w
rite
an
exam
ple
of
wh
y y
ou
dis
agre
e.
6 ST
EP
1A
, D
, o
r N
SS
tate
men
tS
TE
P 2
A o
r D
1
. A
tri
an
gle
has
no d
iagon
als
.
2
. A
dia
gon
al
of
a p
oly
gon
is
a s
egm
en
t jo
inin
g t
he
mid
poin
ts o
f tw
o s
ides
of
the p
oly
gon
.
3
. T
he s
um
of
the m
easu
res
of
the a
ngle
s in
a p
oly
gon
can
be d
ete
rmin
ed
by s
ubtr
act
ing 2
fro
m t
he n
um
ber
of
sid
es
an
d m
ult
iply
ing t
he r
esu
lt b
y 1
80.
4
. F
or
a q
uad
rila
tera
l to
be a
para
llelo
gra
m i
t m
ust
have
two p
air
s of
para
llel
sid
es.
5
. T
he d
iagon
als
of
a p
ara
llelo
gra
m a
re c
on
gru
en
t.
6
. If
you
kn
ow
th
at
on
e p
air
of
op
posi
te s
ides
of
a
qu
ad
rila
tera
l is
both
para
llel
an
d c
on
gru
en
t, t
hen
you
k
now
th
e q
uad
rila
tera
l is
a p
ara
llelo
gra
m.
7
. If
a q
uad
rila
tera
l is
a r
ect
an
gle
, th
en
all
fou
r an
gle
s are
co
ngru
en
t.
8
. T
he d
iagon
als
of
a r
hom
bu
s are
con
gru
en
t.
9
. T
he p
rop
ert
ies
of
a r
hom
bu
s are
not
tru
e f
or
a s
qu
are
.
10
. A
tra
pezoid
has
on
ly o
ne p
air
of
para
llel
sid
es.
11
. T
he m
ed
ian
of
a t
rap
ezoid
is
perp
en
dic
ula
r to
th
e b
ase
s.
12
. A
n i
sosc
ele
s tr
ap
ezoid
has
exact
ly o
ne p
air
of
con
gru
en
t si
des.
Ste
p 2
Ste
p 1
A D A A D A A D D A D A
Lesson 6-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
5
Gle
nc
oe
Ge
om
etr
y
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
An
gle
s o
f P
oly
go
ns
Po
lyg
on
In
teri
or
An
gle
s S
um
T
he s
egm
en
ts t
hat
con
nect
th
e n
on
con
secu
tive
vert
ices
of
a p
oly
gon
are
call
ed
dia
go
na
ls.
Dra
win
g a
ll o
f th
e d
iagon
als
fr
om
on
e v
ert
ex o
f an
n-g
on
sep
ara
tes
the p
oly
gon
in
to n
- 2
tri
an
gle
s. T
he s
um
of
the m
easu
res
of
the i
nte
rior
an
gle
s of
the p
oly
gon
can
be f
ou
nd
by a
dd
ing t
he m
easu
res
of
the i
nte
rior
an
gle
s of
those
n -
2 t
rian
gle
s.
A
co
nv
ex
po
lyg
on
ha
s
13
sid
es.
Fin
d t
he
su
m o
f th
e m
ea
su
re
s
of
the
in
terio
r a
ng
les.
(n
- 2
)
180 =
(13 -
2)
180
= 1
1
180
= 1
980
T
he
me
asu
re
of
an
inte
rio
r a
ng
le o
f a
re
gu
lar p
oly
go
n i
s
12
0.
Fin
d t
he
nu
mb
er o
f sid
es.
Th
e n
um
ber
of
sid
es
is n
, so
th
e s
um
of
the
measu
res
of
the i
nte
rior
an
gle
s is
120n
.
12
0n
= (
n -
2)
180
12
0n
= 1
80n
- 3
60
-60
n =
-360
n
= 6
Exerc
ises
Fin
d t
he
su
m o
f th
e m
ea
su
re
s o
f th
e i
nte
rio
r a
ng
les o
f e
ach
co
nv
ex
po
lyg
on
.
1
. d
eca
gon
2. 16-g
on
3. 30-g
on
4. oct
agon
5. 12-g
on
6. 35-g
on
Th
e m
ea
su
re
of
an
in
terio
r a
ng
le o
f a
re
gu
lar p
oly
go
n i
s g
ive
n.
Fin
d t
he
nu
mb
er
of
sid
es i
n t
he
po
lyg
on
.
7
. 150
8. 160
9. 175
10
. 165
11
. 144
12
. 135
13. F
ind
th
e v
alu
e o
f x.
6 -1
7x°
( 4x+
10) °
( 4x+
5) ° ( 6
x+
10) °
( 5x-
5) °
AB
C
D
E
Po
lyg
on
In
teri
or
An
gle
Su
m T
he
ore
mT
he
su
m o
f th
e in
terio
r a
ng
le m
ea
su
res o
f a
n n
-sid
ed
co
nve
x p
oly
go
n is (
n -
2)
· 1
80
.
Exam
ple
1Exam
ple
2
1
440
2520
5040
1
080
1800
5940
1
2
18
72
2
4
10
8
2
0
Answers (Anticipation Guide and Lesson 6-1)
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Chapter 6 A2 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
6
Gle
nc
oe
Ge
om
etr
y
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
An
gle
s o
f P
oly
go
ns
Po
lyg
on
Ex
teri
or
An
gle
s S
um
T
here
is
a s
imp
le r
ela
tion
ship
am
on
g
the e
xte
rior
an
gle
s of
a c
on
vex p
oly
gon
.
F
ind
th
e s
um
of
the
me
asu
re
s o
f th
e e
xte
rio
r a
ng
les,
on
e a
t
ea
ch
ve
rte
x,
of
a c
on
ve
x 2
7-g
on
.
For
an
y c
on
vex p
oly
gon
, th
e s
um
of
the m
easu
res
of
its
exte
rior
an
gle
s, o
ne a
t each
vert
ex,
is 3
60.
F
ind
th
e m
ea
su
re
of
ea
ch
ex
terio
r a
ng
le o
f
re
gu
lar h
ex
ag
on
ABCDEF
.
Th
e s
um
of
the m
easu
res
of
the e
xte
rior
an
gle
s is
360 a
nd
a h
exagon
has
6 a
ngle
s. I
f n
is
the m
easu
re o
f each
exte
rior
an
gle
, th
en
6n
= 3
60
n
= 6
0
Th
e m
easu
re o
f each
exte
rior
an
gle
of
a r
egu
lar
hexagon
is
60.
Exerc
ises
Fin
d t
he
su
m o
f th
e m
ea
su
re
s o
f th
e e
xte
rio
r a
ng
les o
f e
ach
co
nv
ex
po
lyg
on
.
1
. d
eca
gon
2. 16-g
on
3. 36-g
on
Fin
d t
he
me
asu
re
of
ea
ch
ex
terio
r a
ng
le f
or e
ach
re
gu
lar p
oly
go
n.
4. 12-g
on
5. h
exagon
6. 20-g
on
7
. 40-g
on
8. h
ep
tagon
9. 12-g
on
10
. 24-g
on
1
1. d
od
eca
gon
1
2. oct
agon
AB
C
DE
F
6 -1
Po
lyg
on
Ex
teri
or
An
gle
Su
m T
he
ore
m
Th
e s
um
of
the
exte
rio
r a
ng
le m
ea
su
res o
f a
co
nve
x p
oly
go
n,
on
e a
ng
le
at
ea
ch
ve
rte
x,
is 3
60
.
Exam
ple
1
Exam
ple
2
3
60
360
360
3
0
60
18
9
5
1.4
3
0
1
5
30
45
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 6-1
Ch
ap
ter
6
7
Gle
nc
oe
Ge
om
etr
y
Sk
ills
Pra
ctic
e
An
gle
s o
f P
oly
go
ns
Fin
d t
he
su
m o
f th
e m
ea
su
re
s o
f th
e i
nte
rio
r a
ng
les o
f e
ach
co
nv
ex
po
lyg
on
.
1
. n
on
agon
2. h
ep
tagon
3. d
eca
gon
Th
e m
ea
su
re
of
an
in
terio
r a
ng
le o
f a
re
gu
lar p
oly
go
n i
s g
ive
n.
Fin
d t
he
nu
mb
er o
f sid
es i
n t
he
po
lyg
on
.
4. 108
5. 120
6. 150
Fin
d t
he
me
asu
re
of
ea
ch
in
terio
r a
ng
le.
7
.
8.
9
.
10
.
Fin
d t
he
me
asu
re
s o
f e
ach
in
terio
r a
ng
le o
f e
ach
re
gu
lar p
oly
go
n.
11
. qu
ad
rila
tera
l 1
2. p
en
tagon
1
3. d
od
eca
gon
Fin
d t
he
me
asu
re
s o
f e
ach
ex
terio
r a
ng
le o
f e
ach
re
gu
lar p
oly
go
n.
14
. oct
agon
15
. n
on
agon
1
6. 12-g
on
6-1
x°
x°
( 2x-
15) °
( 2x-
15) °
AB
CD
( 2x) °
( 2x+
20) °
( 3x-
10) °
( 2x-
10) °
LM
NP
(2x+
16)°
(x+
14)°
(x+
14
)°
(2x+
16
)°(7
x)°
(7x)°
(4x)°
(4x)°
(7x)°
(7x)°
1
260
900
1440
5
6
1
2
m
∠A
= 1
15,
m∠
B =
65,
m
∠L =
100,
m∠
M =
110,
m∠
C =
115,
m∠
D =
65
m
∠N
= 7
0,
m∠
P =
80
m
∠S
= 1
16,
m∠
T =
116,
m∠
D =
140,
m∠
E =
140,
m
∠W
= 6
4,
m∠
U =
64
m∠
I =
80,
m∠
F =
80,
m∠
H =
140,
m∠
G =
140
9
0,
90
108,
72
150,
30
4
5
40
30
Answers (Lesson 6-1)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Chapter 6 A3 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
8
Gle
nc
oe
Ge
om
etr
y
Fin
d t
he
su
m o
f th
e m
ea
su
re
s o
f th
e i
nte
rio
r a
ng
les o
f e
ach
co
nv
ex
po
lyg
on
.
1
. 11-g
on
2. 14-g
on
3. 17-g
on
Th
e m
ea
su
re
of
an
in
terio
r a
ng
le o
f a
re
gu
lar p
oly
go
n i
s g
ive
n.
Fin
d t
he
nu
mb
er o
f sid
es i
n t
he
po
lyg
on
.
4. 144
5. 156
6. 160
Fin
d t
he
me
asu
re
of
ea
ch
in
terio
r a
ng
le.
7
.
8.
Fin
d t
he
me
asu
re
s o
f a
n e
xte
rio
r a
ng
le a
nd
an
in
terio
r a
ng
le g
ive
n t
he
nu
mb
er o
f
sid
es o
f e
ach
re
gu
lar p
oly
go
n.
Ro
un
d t
o t
he
ne
are
st
ten
th,
if n
ece
ssa
ry
.
9
. 16
10
. 24
11
. 30
12
. 14
13
. 22
14
. 40
15
. C
RY
ST
ALLO
GR
AP
HY
C
ryst
als
are
cla
ssif
ied
acc
ord
ing t
o s
even
cry
stal
syst
em
s. T
he
basi
s of
the c
lass
ific
ati
on
is
the s
hap
es
of
the f
ace
s of
the c
ryst
al.
Tu
rqu
ois
e b
elo
ngs
to
the t
ricl
inic
syst
em
. E
ach
of
the s
ix f
ace
s of
turq
uois
e i
s in
th
e s
hap
e o
f a p
ara
llelo
gra
m.
Fin
d t
he s
um
of
the m
easu
res
of
the i
nte
rior
an
gle
s of
on
e s
uch
face
.
Practi
ce
An
gle
s o
f P
oly
go
ns
6 -1
x°
( 2x+
15
) °( 3
x-
20
) °
( x+
15
) °
JK
MN
(6x-
4)°
(2x+
8)°
(6x-
4)°
(2x+
8)°
m
∠R
= 1
28,
m∠
S =
52
m∠
T =
128,
m∠
U =
52
1
620
2160
2700
1
0
15
18
m
∠J =
115,
m∠
K =
130,
m∠
M =
50,
m∠
N =
65
1
57.5
, 22.5
1
65,
15
168,
12
1
54.3
, 25.7
1
63.6
, 16.4
1
71,
9
3
60
Lesson 6-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
9
Gle
nc
oe
Ge
om
etr
y
Wo
rd
Pro
ble
m P
racti
ce
An
gle
s o
f P
oly
go
ns
La
Trib
un
a
1
. A
RC
HIT
EC
TU
RE
In
th
e U
ffiz
i gall
ery
in
Flo
ren
ce,
Italy
, th
ere
is
a r
oom
bu
ilt
by B
uon
tale
nti
call
ed
th
e T
ribu
ne (
La
Trib
un
a i
n I
tali
an
). T
his
room
is
shap
ed
li
ke a
regu
lar
oct
agon
.
Wh
at
an
gle
do c
on
secu
tive w
all
s of
the
Tri
bu
ne m
ak
e w
ith
each
oth
er?
2
. B
OX
ES
Jasm
ine i
s d
esi
gn
ing b
oxes
she
wil
l u
se t
o s
hip
her
jew
elr
y.
Sh
e w
an
ts
to s
hap
e t
he b
ox l
ike a
regu
lar
poly
gon
. In
ord
er
for
the b
oxes
to p
ack
tig
htl
y,
she d
eci
des
to u
se a
regu
lar
poly
gon
th
at
has
the p
rop
ert
y t
hat
the m
easu
re o
f it
s in
teri
or
an
gle
s is
half
th
e m
easu
re o
f it
s exte
rior
an
gle
s. W
hat
regu
lar
poly
gon
sh
ou
ld s
he u
se?
3
. T
HE
AT
ER
A
th
eate
r fl
oor
pla
n i
s sh
ow
n
in t
he f
igu
re.
Th
e u
pp
er
five s
ides
are
p
art
of
a r
egu
lar
dod
eca
gon
.
Fin
d m∠
1.
4. A
RC
HE
OLO
GY
A
rch
eolo
gis
ts u
neart
hed
p
art
s of
two a
dja
cen
t w
all
s of
an
an
cien
t ca
stle
.
Befo
re i
t w
as
un
eart
hed
, th
ey k
new
fro
m
an
cien
t te
xts
th
at
the c
ast
le w
as
shap
ed
li
ke a
regu
lar
poly
gon
, bu
t n
obod
y k
new
h
ow
man
y s
ides
it h
ad
. S
om
e s
aid
6,
oth
ers
8,
an
d s
om
e e
ven
said
100.
Fro
m
the i
nfo
rmati
on
in
th
e f
igu
re,
how
man
y
sid
es
did
th
e c
ast
le r
eall
y h
ave?
5
. P
OLY
GO
N P
AT
HIn
Ms.
Ric
kets
’ m
ath
cl
ass
, st
ud
en
ts m
ad
e a
“p
oly
gon
path
” th
at
con
sist
s of
regu
lar
poly
gon
s of
3,
4,
5,
an
d 6
sid
es
join
ed
togeth
er
as
show
n.
a.
Fin
d m∠
2 a
nd
m∠
5.
b.
Fin
d m∠
3 a
nd
m∠
4.
c.
Wh
at
is m∠
1?
6 -1
21
3
4
5
stag
e
1
24˚
1
35°
a
n e
qu
ilate
ral
tria
ng
le
1
20
1
5
90 a
nd
60
162 a
nd
132
96
Answers (Lesson 6-1)
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Chapter 6 A4 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
10
G
len
co
e G
eo
me
try
En
rich
men
t
Cen
tral
An
gle
s o
f R
eg
ula
r P
oly
go
ns
You
have l
earn
ed
abou
t th
e i
nte
rior
an
d e
xte
rior
an
gle
s of
a p
oly
gon
. R
egu
lar
poly
gon
s als
o h
ave c
en
tra
l a
ng
les.
A c
en
tral
an
gle
is
measu
red
fro
m t
he c
en
ter
of
the p
oly
gon
.
Th
e c
en
ter
of
a p
oly
gon
is
the p
oin
t equ
idis
tan
t fr
om
all
of
th
e v
ert
ices
of
the p
oly
gon
, ju
st a
s th
e c
en
ter
of
a c
ircl
e i
s th
e p
oin
t equ
idis
tan
t fr
om
all
of
the p
oin
ts o
n t
he c
ircl
e.
Th
e
cen
tral
an
gle
is
the a
ngle
dra
wn
wit
h t
he v
ert
ex a
t th
e
cen
ter
of
the c
ircl
e a
nd
th
e s
ides
of
an
gle
dra
wn
th
rou
gh
co
nse
cuti
ve v
ert
ices
of
the p
oly
gon
. O
ne o
f th
e c
en
tral
an
gle
s th
at
can
be d
raw
n i
n t
his
regu
lar
hexagon
is
∠A
PB
. Y
ou
may r
em
em
ber
from
mak
ing c
ircl
e g
rap
hs
that
there
are
360°
aro
un
d t
he c
en
ter
of
a c
ircl
e.
1. B
y u
sin
g l
ogic
or
by d
raw
ing s
ketc
hes,
fin
d t
he m
easu
re o
f th
e c
en
tral
an
gle
of
each
re
gu
lar
poly
gon
.
2. M
ak
e a
con
ject
ure
abou
t h
ow
th
e m
easu
re o
f a c
en
tral
an
gle
of
a r
egu
lar
poly
gon
rela
tes
to t
he m
easu
res
of
the i
nte
rior
an
gle
s an
d e
xte
rior
an
gle
s of
a r
egu
lar
poly
gon
.
3
. C
HA
LLE
NG
E In
obtu
se △
AB
C,
−−
−
BC
is
the l
on
gest
sid
e.
−−
AC
is
als
o a
sid
e o
f a
21-s
ided
regu
lar
poly
gon
. −
−
AB
is
als
o a
sid
e o
f a 2
8-s
ided
regu
lar
poly
gon
. T
he
21-s
ided
regu
lar
poly
gon
an
d t
he 2
8-s
ided
regu
lar
poly
gon
have t
he s
am
e
cen
ter
poin
t P
. F
ind
n i
f −
−−
BC
is
a s
ide o
f a n
-sid
ed
regu
lar
poly
gon
th
at
has
cen
ter
poin
t P
.
(H
int:
Sk
etc
h a
cir
cle w
ith
cen
ter
P a
nd
pla
ce p
oin
ts A
, B
, an
d C
on
th
e c
ircl
e.)
6 -1
A
B
P
r
1
20
9
0
72
6
0
ab
ou
t 51.4
3
4
5
T
he m
easu
re o
f th
e e
xte
rio
r an
gle
eq
uals
th
e m
easu
re o
f th
e c
en
tral
an
gle
. T
he c
en
tral
an
gle
is s
up
ple
men
tary
to
in
teri
or
an
gle
.
n
= 1
2
Lesson 6-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
11
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Para
llelo
gra
ms
Sid
es
an
d A
ng
les
of
Pa
rall
elo
gra
ms
A q
uad
rila
tera
l w
ith
both
pair
s of
op
posi
te s
ides
para
llel
is a
pa
ra
lle
log
ra
m.
Here
are
fou
r im
port
an
t p
rop
ert
ies
of
para
llelo
gra
ms.
If
ABCD
is a
pa
ra
lle
log
ra
m,
fin
d t
he
va
lue
of
ea
ch
va
ria
ble
.−
−
AB
an
d −
−−
CD
are
op
posi
te s
ides,
so −
−
AB
$ −
−−
CD
.
2a
= 3
4
a
= 1
7
∠A
an
d ∠
C a
re o
pp
osi
te a
ngle
s, s
o ∠
A $
∠C
.
8b =
112
b =
14
Exerc
ises
Fin
d t
he
va
lue
of
ea
ch
va
ria
ble
.
1
.
2.
3
.
4.
5
.
6.
PQ
RS
BA
DC
34
2a
8b°
112°
4y°
3x°
8y
88
6x°
72
x
30
x150
2y
6 -2
If P
QR
S i
s a
pa
rall
elo
gra
m,
the
n
If a
qu
ad
rila
tera
l is
a p
ara
llelo
gra
m,
the
n its
op
po
site
sid
es a
re c
on
gru
en
t.
−−
−
PQ
$ −
−
SR
an
d −
−
PS
$ −
−−
QR
If a
qu
ad
rila
tera
l is
a p
ara
llelo
gra
m,
the
n its
op
po
site
an
gle
s a
re c
on
gru
en
t.
∠P
$ ∠
R a
nd
∠S
$ ∠
Q
If a
qu
ad
rila
tera
l is
a p
ara
llelo
gra
m,
the
n its
co
nse
cu
tive
an
gle
s a
re
su
pp
lem
en
tary
.
∠P
an
d ∠
S a
re s
up
ple
me
nta
ry;
∠S
an
d ∠
R a
re s
up
ple
me
nta
ry;
∠R
an
d ∠
Q a
re s
up
ple
me
nta
ry;
∠Q
an
d ∠
P a
re s
up
ple
me
nta
ry.
If a
pa
ralle
log
ram
ha
s o
ne
rig
ht
an
gle
,
the
n it
ha
s f
ou
r rig
ht
an
gle
s.
If m
∠P
= 9
0,
the
n m
∠Q
= 9
0,
m∠
R =
90
, a
nd
m∠
S =
90
.
Exam
ple
3y
12
6x 60°55°
5x°
2y°
6x°
12
x°
3y°
x
= 3
0;
y =
22.5
x =
15;
y =
11
x
= 2
; y =
4
x =
10;
y =
40
x
= 1
3;
y =
32.5
x =
5;
y =
180
Answers (Lesson 6-1 and Lesson 6-2)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Chapter 6 A5 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
12
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Para
llelo
gra
ms
Dia
go
na
ls o
f P
ara
lle
log
ram
s T
wo i
mp
ort
an
t p
rop
ert
ies
of
p
ara
llelo
gra
ms
deal
wit
h t
heir
dia
gon
als
.
F
ind
th
e v
alu
e o
f x a
nd
y i
n p
ara
lle
log
ra
m ABCD
.
Th
e d
iagon
als
bis
ect
each
oth
er,
so A
E =
CE
an
d D
E =
BE
.
6x =
24
4y =
18
x =
4
y =
4.5
Exerc
ises
Fin
d t
he
va
lue
of
ea
ch
va
ria
ble
.
1
.
2.
3.
4
.
5.
6.
CO
OR
DIN
AT
E G
EO
ME
TR
Y
Fin
d t
he
co
ord
ina
tes o
f th
e i
nte
rse
cti
on
of
the
d
iag
on
als
of
ABCD
wit
h t
he
giv
en
ve
rti
ce
s.
7. A
(3,
6),
B(5
, 8),
C(3
, −
2),
an
d D
(1,
−4)
8. A
(−4,
3),
B(2
, 3),
C(−
1,
−2),
an
d D
(−7,
−2)
9
. P
RO
OF W
rite
a p
ara
gra
ph
pro
of
of
the f
oll
ow
ing.
G
ive
n:
!A
BC
D
P
ro
ve
: △
AE
D #
△B
EC
AB
CD
P
AB
E
CD
6x
18
24
4y
3x
4y
812
4x
2y
28
3x
30
°10
y
AB
CD
E
2x°6
0°
4y°
3x°
2y
12
6 -2
If A
BC
D i
s a
pa
rall
elo
gra
m,
the
n
If a
qu
ad
rila
tera
l is
a p
ara
llelo
gra
m,
the
n
its d
iag
on
als
bis
ect
ea
ch
oth
er.
AP
= P
C a
nd
DP
= P
B
If a
qu
ad
rila
tera
l is
a p
ara
llelo
gra
m,
the
n e
ach
dia
go
na
l se
pa
rate
s t
he
pa
ralle
log
ram
into
tw
o c
on
gru
en
t tr
ian
gle
s.
△A
CD
# △
CA
B a
nd
△A
DB
# △
CB
D
Exam
ple
x
17
4y
x
= 4
; y =
2
x =
7;
y =
14
x =
15;
y =
7.5
x
= 3
1
−
3 ;
y =
10 √
$
3
x =
15;
y =
6 √
$
2
x =
15;
y =
√ $
$
241
D
iag
on
als
of
a p
ara
lle
log
ram
bis
ec
t e
ac
h o
the
r, s
o −
−
AE
& −
−
CE
an
d −
−
BE
& −
−
DE
. O
pp
os
ite
sid
es
of
a p
ara
lle
log
ram
are
co
ng
rue
nt,
th
ere
fore
−−
AD
& −
−
BC
. B
ec
au
se
co
rre
sp
on
din
g p
art
s o
f th
e t
wo
tri
an
gle
s a
re c
on
gru
en
t, t
he
tr
ian
gle
s a
re c
on
gru
en
t b
y S
SS
.
(
3,
2)
(-
2.5
, 0.5
)
Lesson 6-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
13
G
len
co
e G
eo
me
try
Sk
ills
Pra
ctic
e
Para
llelo
gra
ms
ALG
EB
RA
Fin
d t
he
va
lue
of
ea
ch
va
ria
ble
.
1
.
2.
3
.
4.
5
.
6.
CO
OR
DIN
AT
E G
EO
ME
TR
Y F
ind
th
e c
oo
rd
ina
tes o
f th
e i
nte
rse
cti
on
of
the
dia
go
na
ls
of
HJKL
wit
h t
he
giv
en
ve
rti
ce
s.
7
. H
(1,
1),
J(2
, 3),
K(6
, 3),
L(5
, 1)
8. H
(-1,
4),
J(3
, 3),
K(3
, -
2),
L(-
1,
-1)
9. P
RO
OF
W
rite
a p
ara
gra
ph
pro
of
of
the t
heore
m C
on
secu
tive
an
gle
s in
a p
ara
llel
ogra
m
are
su
pp
lem
enta
ry.
DC
BA
6-2
x-
2y
+1
1926
x°44°
y°
4x- 2
3y-
1
y+
5
x + 10
2b
-1
b+
3
3a
-5
2a
a+
14
6a
+4
10b
+19b
+8
(x+
8)°
(3x)°
(y+
9)°
a
= 5
, b
= 4
x =
136,
y =
44
(
3.5
, 2)
(1
, 1)
G
iven
:
AB
CD
P
rove:
∠A
an
d ∠
B a
re s
up
ple
men
tary
.∠
B a
nd
∠C
are
su
pp
lem
en
tary
.∠
C a
nd
∠D
are
su
pp
lem
en
tary
.∠
D a
nd
∠A
are
su
pp
lem
en
tary
.
P
roo
f: W
e a
re g
ive
n
AB
CD
, s
o w
e k
no
w t
ha
t −
−
AB
||
−−
CD
an
d −
−
BC
||
−−
DA
by
the
de
fin
itio
n o
f a
pa
rall
elo
gra
m.
We
als
o k
no
w t
ha
t if
tw
o p
ara
lle
l li
ne
s a
re
cu
t b
y a
tra
ns
ve
rsa
l, t
he
n c
on
se
cu
tiv
e i
nte
rio
r a
ng
les
are
su
pp
lem
en
tary
. S
o,
∠A
an
d ∠
B,
∠B
an
d ∠
C,
∠C
an
d ∠
D,
an
d ∠
D a
nd
∠A
are
pa
irs
of
su
pp
lem
en
tary
an
gle
s.
x
= 4
3,
y =
12
0
x =
4,
y =
3
x
= 2
1,
y =
25
a =
2,
b =
7
Answers (Lesson 6-2)
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Chapter 6 A6 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
14
G
len
co
e G
eo
me
try
Practi
ce
Para
llelo
gra
ms
ALG
EB
RA
Fin
d t
he
va
lue
of e
ach
va
ria
ble
.
1
.
2.
3
.
4.
ALG
EB
RA
U
se
RSTU
to
fin
d e
ach
me
asu
re
or v
alu
e.
5
. m∠RST=
6. m∠STU=
7
. m∠TUR=
8. b=
CO
OR
DIN
AT
E G
EO
ME
TR
Y F
ind
th
e c
oo
rd
ina
tes o
f th
e i
nte
rse
cti
on
of
the
dia
go
na
ls
of PRYZ
wit
h t
he
giv
en
ve
rti
ce
s.
9. P
(2,
5),
R(3
, 3),
Y(-
2, -
3),
Z(-
3, -
1)
10
. P
(2,
3),
R(1
, -
2),
Y(-
5, -
7),
Z(-
4, -
2)
11. P
RO
OF
W
rite
a p
ara
gra
ph
pro
of
of
the f
oll
ow
ing.
G
ive
n: !PRST
an
d !
PQVU
Pro
ve
: ∠V
" ∠S
12
. C
ON
ST
RU
CT
ION
M
r. R
od
riqu
ez u
sed
th
e p
ara
llelo
gra
m a
t th
e r
igh
t to
d
esi
gn
a h
err
ingbon
e p
att
ern
for
a p
avin
g s
ton
e.
He w
ill
use
th
e p
avin
g
ston
e f
or
a s
idew
alk
. If
m∠
1 i
s 130,
fin
d m∠
2, m∠
3,
an
d m∠
4.
TU
SR
B30°
23
4b-
1
25°
PR
S
UVQ
T
12
34
6 -2
(4x)
°
(y+
10)°
(2y-
40)°
4x
5y-
83y+
1
x+6
y+3
15
x-
3
12
b+1
2b
a+2
3a-
4
125
(
0,
1)
(-
1.5
, -
2)
P
roo
f: W
e a
re g
iven
P
RS
T a
nd
P
QV
U.
Sin
ce o
pp
osit
e a
ng
les o
f a
para
llelo
gra
m a
re c
on
gru
en
t, ∠
P #
∠V
an
d ∠
P #
∠S
. S
ince c
on
gru
en
ce
of
an
gle
s i
s t
ran
sit
ive, ∠
V #
∠S
by t
he T
ran
sit
ive P
rop
ert
y o
f
co
ng
ruen
ce.
5
0,
130,
50
a
= 3
, b
= 1
x =
30˚,
y =
50˚
x
= 1
8,
y =
9
x =
2,
y =
4.5
125
6
55
Lesson 6-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
15
G
len
co
e G
eo
me
try
1
. W
ALK
WA
Y A
walk
way i
s m
ad
e b
y
ad
join
ing f
ou
r p
ara
llelo
gra
ms
as
show
n.
Are
th
e e
nd
segm
en
ts a
an
d e
para
llel
to
each
oth
er?
Exp
lain
.
2
. D
IST
AN
CE
F
ou
r fr
ien
ds
live a
t th
e f
ou
r co
rners
of
a b
lock
sh
ap
ed
lik
e a
p
ara
llelo
gra
m.
Gra
cie l
ives
3 m
iles
aw
ay
from
Ken
ny.
How
far
ap
art
do T
ere
sa
an
d T
ravis
liv
e f
rom
each
oth
er?
3. S
OC
CE
R F
ou
r so
ccer
pla
yers
are
loca
ted
at
the c
orn
ers
of
a p
ara
llelo
gra
m.
Tw
o
of
the p
layers
in
op
posi
te c
orn
ers
are
th
e g
oali
es.
In
ord
er
for
goali
e A
to b
e
able
to s
ee t
he t
hre
e o
thers
, sh
e m
ust
be a
ble
to s
ee a
cert
ain
an
gle
x i
n h
er
field
of
vis
ion
.
Wh
at
an
gle
does
the o
ther
goali
e h
ave t
o
be a
ble
to s
ee i
n o
rder
to k
eep
an
eye o
n
the o
ther
thre
e p
layers
?
4
. V
EN
N D
IAG
RA
MS
M
ak
e a
Ven
n
dia
gra
m s
how
ing t
he r
ela
tion
ship
betw
een
squ
are
s, r
ect
an
gle
s, a
nd
p
ara
llelo
gra
ms.
5
. S
KY
SC
RA
PE
RS
O
n v
aca
tion
, T
on
y’s
fa
mil
y t
ook
a h
eli
cop
ter
tou
r of
the c
ity.
Th
e p
ilot
said
th
e
new
est
bu
ild
ing
in t
he c
ity w
as
the b
uil
din
g w
ith
th
is t
op
vie
w.
He
told
Ton
y t
hat
the
exte
rior
an
gle
by
the f
ron
t en
tran
ce
is 7
2°.
Ton
y
wan
ted
to k
now
m
ore
abou
t th
e b
uil
din
g,
so h
e d
rew
th
is
dia
gra
m a
nd
use
d h
is g
eom
etr
y s
kil
ls t
o
learn
a f
ew
more
th
ings.
Th
e f
ron
t en
tran
ce i
s n
ext
to v
ert
ex B
.
a.
Wh
at
are
th
e m
easu
res
of
the f
ou
r an
gle
s of
the p
ara
llelo
gra
m?
b.
How
man
y p
air
s of
con
gru
en
t tr
ian
gle
s are
th
ere
in
th
e f
igu
re?
Wh
at
are
th
ey?
Wo
rd
Pro
ble
m P
racti
ce
Para
llelo
gra
ms
AB
CD
E
72˚
par
alle
logra
ms
rect
ang
les
squ
ares
Goal
ie B
Pla
yer
A
Go
alie
A
x
Pla
yer
B
Tere
saTr
avis
Gra
cie
Ken
ny
a
e
6 -2
Y
es.
Op
po
sit
e s
ides o
f a
para
llelo
gra
m a
re p
ara
llel
an
d t
he
para
llel
pro
pert
y i
s t
ran
sit
ive.
3
mi
H
e o
r sh
e a
lso
has t
o s
ee
an
gle
x.
72,
72,
108,
108
4 p
air
s;
△A
BE
# △
DC
E,
△A
BC
# △
DC
B,
△A
CE
# △
DB
E,
an
d
△A
BD
# △
DC
A
Answers (Lesson 6-2)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Chapter 6 A7 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
16
G
len
co
e G
eo
me
try
En
rich
men
t
Dia
go
nals
of
Para
llelo
gra
ms
In s
om
e d
raw
ings
the d
iagon
al
of
a p
ara
llelo
gra
m a
pp
ears
to b
e t
he a
ngle
bis
ect
or
of
both
op
posi
te a
ngle
s. W
hen
mig
ht
that
be t
rue?
1. G
ive
n:
Para
llelo
gra
m P
QR
S w
ith
dia
gon
al
−−
PR
.
−−
PR
is
an
an
gle
bis
ect
or
of
∠Q
PS
an
d ∠
QR
S.
W
hat
typ
e o
f p
ara
llelo
gra
m i
s P
QR
S?
Ju
stif
y
you
r an
swer.
2. G
ive
n:
Para
llelo
gra
m W
PR
K w
ith
an
gle
bis
ect
or
−−
−
KD
, D
P =
5,
an
d W
D =
7.
F
ind
WK
an
d K
R.
3. R
efe
r to
Exerc
ise 2
. W
rite
a s
tate
men
t abou
t p
ara
llelo
gra
m W
PR
K a
nd
an
gle
bis
ect
or
−−
−
KD
.
4. G
ive
n:
Para
llelo
gra
m A
BC
D w
ith
d
iagon
al
−−
−
BD
an
d a
ngle
bis
ect
or
−−
BP
. P
D =
5,
BP
= 6
, an
d C
P =
6.
Th
e p
eri
mete
r of
tria
ngle
PC
D i
s 15.
F
ind
AB
an
d B
C.
P
QR
S
PD
A
BC
PD
R
K
W
6 -2
∠
QR
S !
∠Q
PS
op
po
sit
e a
ng
les o
f a
para
llelo
gra
m
∠
QP
R !
∠R
PS
defi
nit
ion
of
an
gle
bis
ecto
r
∠
QR
P !
∠S
RP
defi
nit
ion
of
an
gle
bis
ecto
r
∠
QP
R !
∠P
RS
alt
ern
ate
in
teri
or
an
gle
s
∠
RP
S !
∠P
RS
Tra
nsit
ive P
rop
ert
y
−−
SP
! −
−
SR
Is
oscele
s T
rian
gle
Th
eo
rem
S
o a
ll s
ides a
re !
an
d Q
RP
S i
s a
rh
om
bu
s
W
K =
7,
KR
= 1
2
S
am
ple
an
sw
er:
Para
llelo
gra
m
WK
RP
is n
ot
a r
ho
mb
us a
nd
−−
KD
is n
ot
a d
iag
on
al.
A
B =
4,
BC
= 9
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 6-3
Ch
ap
ter
6
17
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Tests
fo
r P
ara
llelo
gra
ms
Co
nd
itio
ns
for
Pa
rall
elo
gra
ms
Th
ere
are
man
y w
ays
to
est
abli
sh t
hat
a q
uad
rila
tera
l is
a p
ara
llelo
gra
m.
F
ind
x a
nd
y s
o t
ha
t FGHJ
is a
pa
ra
lle
log
ra
m.
FG
HJ
is
a p
ara
llelo
gra
m i
f th
e l
en
gth
s of
the o
pp
osi
te
sid
es
are
equ
al.
6x +
3 =
15
4x -
2y =
2
6x =
12
4(2
) -
2y =
2
x =
2
8 -
2y =
2
-2y =
-6
y =
3
Exerc
ises
Fin
d x
an
d y
so
th
at
the
qu
ad
ril
ate
ra
l is
a p
ara
lle
log
ra
m.
1
.
2.
3
.
4.
5
.
6.
AB
CD
E
JHG
F6
x+
3
15
4x
- 2
y2
2x
-2
2y
8
12
11
x°
5y°
55
°
5x
°5
y°
25
°
( x+
y) °
2x
°
30
°
24
°
6y
°
3x
°
9x
°6
y4
5°
18
6 -3
If:
If:
bo
th p
airs o
f o
pp
osite
sid
es a
re p
ara
llel,
−−
AB
||
−−−
DC
an
d −
−
AD
||
−−−
BC
,
bo
th p
airs o
f o
pp
osite
sid
es a
re c
on
gru
en
t, −
−
AB
# −−
− D
C a
nd
−−
AD
# −−
− B
C ,
bo
th p
airs o
f o
pp
osite
an
gle
s a
re c
on
gru
en
t,∠
AB
C #
∠A
DC
an
d ∠
DA
B #
∠B
CD
,
the
dia
go
na
ls b
ise
ct
ea
ch
oth
er,
−−
AE
# −
−
CE
an
d −
−
DE
# −
−
BE
,
on
e p
air o
f o
pp
osite
sid
es is c
on
gru
en
t a
nd
pa
ralle
l, −
−
AB
||
−−−
CD
an
d −
−
AB
# −−
− C
D ,
or
−−
AD
||
−−−
BC
an
d −
−
AD
# −−
− B
C ,
the
n:
the
fig
ure
is a
pa
ralle
log
ram
.th
en
: A
BC
D is a
pa
ralle
log
ram
.
Exam
ple
x =
7;
y =
4x =
5;
y =
25
x =
31;
y =
5x =
5;
y =
3
x =
15;
y =
9x =
30;
y =
15
Answers (Lesson 6-2 and Lesson 6-3)
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Chapter 6 A8 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
18
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Tests
fo
r P
ara
llelo
gra
ms
Pa
rall
elo
gra
ms
on
th
e C
oo
rdin
ate
Pla
ne
O
n t
he c
oord
inate
pla
ne,
the D
ista
nce
, S
lop
e,
an
d M
idp
oin
t F
orm
ula
s ca
n b
e u
sed
to t
est
if
a q
uad
rila
tera
l is
a p
ara
llelo
gra
m.
D
ete
rm
ine
wh
eth
er A
BC
D i
s a
pa
ra
lle
log
ra
m.
Th
e v
ert
ices
are
A(-
2,
3),
B(3
, 2),
C(2
, -
1),
an
d D
(-3,
0).
Meth
od
1:
Use
th
e S
lop
e F
orm
ula
, m
= y
2 -
y1
−
x2 -
x1
.
slop
e o
f −−
− A
D =
3 -
0
−
-2 -
(-
3)
= 3
−
1 =
3
slop
e o
f −−
− B
C =
2 -
(-
1)
−
3 -
2
=
3
−
1 =
3
slop
e o
f −
−
AB
=
2 -
3
−
3 -
(-
2)
= -
1
−
5
slop
e o
f −−
− C
D =
-
1 -
0
−
2 -
(-
3)
= -
1
−
5
Sin
ce o
pp
osi
te s
ides
have t
he s
am
e s
lop
e,
−−
AB
||
−−−
CD
an
d −−
− A
D |
| −−
− B
C .
Th
ere
fore
, A
BC
D i
s a
para
llelo
gra
m b
y d
efi
nit
ion
.
Meth
od
2:
Use
th
e D
ista
nce
Form
ula
, d
= √
##
##
##
##
#
(x
2 -
x1)2
+ (
y2 -
y1)2
.
AB
= √
##
##
##
##
#
(-
2 -
3)2
+ (
3 -
2)2
= √
##
#
25 +
1 o
r √
##
26
CD
= √
##
##
##
##
##
(2
- (
-3))
2 +
(-
1 -
0)2
= √
##
#
25 +
1 o
r √
##
26
AD
= √
##
##
##
##
##
(-
2 -
(-
3))
2 +
(3 -
0)2
= √
##
#
1 +
9 o
r √
##
10
BC
= √
##
##
##
##
#
(3
- 2
)2 +
(2 -
(-
1))
2 =
√ #
##
1 +
9 o
r √
##
10
Sin
ce b
oth
pair
s of
op
posi
te s
ides
have t
he s
am
e l
en
gth
, −
−
AB
$ −−
− C
D a
nd
−−−
AD
$ −−
− B
C .
Th
ere
fore
, A
BC
D i
s a p
ara
llelo
gra
m b
y T
heore
m 6
.9.
Exerc
ises
Gra
ph
ea
ch
qu
ad
ril
ate
ra
l w
ith
th
e g
ive
n v
erti
ce
s.
De
term
ine
wh
eth
er t
he
fig
ure
is
a p
ara
lle
log
ra
m.
Ju
sti
fy y
ou
r a
nsw
er w
ith
th
e m
eth
od
in
dic
ate
d.
1
. A
(0,
0),
B(1
, 3),
C(5
, 3),
D(4
, 0);
2
. D
(-1,
1),
E(2
, 4),
F(6
, 4),
G(3
, 1);
Slo
pe F
orm
ula
S
lop
e F
orm
ula
3. R
(-1,
0),
S(3
, 0),
T(2
, -
3),
U(-
3,
-2);
4. A
(-3,
2),
B(-
1,
4),
C(2
, 1),
D(0
, -
1);
Dis
tan
ce F
orm
ula
D
ista
nce
an
d S
lop
e F
orm
ula
s
5
. S
(-2,
4),
T(-
1,
-1),
U(3
, -
4),
V(2
, 1);
6
. F
(3,
3),
G(1
, 2),
H(-
3,
1),
I(-
1,
4);
Dis
tan
ce a
nd
Slo
pe F
orm
ula
s M
idp
oin
t F
orm
ula
7. A
para
llelo
gra
m h
as
vert
ices
R(-
2,
-1),
S(2
, 1),
an
d T
(0,
-3).
Fin
d a
ll p
oss
ible
co
ord
inate
s fo
r th
e f
ou
rth
vert
ex.
x
y
O
C
A
D
B
6 -3 Exam
ple
y
es
yes
n
o
yes
y
es
n
o
(
4,
-1),
(0,
3),
or
(-4,
-5)
S
ee s
tud
en
ts’
wo
rk
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 6-3
Ch
ap
ter
6
19
G
len
co
e G
eo
me
try
Sk
ills
Pra
ctic
e
Tests
fo
r P
ara
llelo
gra
ms
De
term
ine
wh
eth
er e
ach
qu
ad
ril
ate
ra
l is
a p
ara
lle
log
ra
m.
Ju
sti
fy y
ou
r a
nsw
er.
1.
2.
3.
4.
CO
OR
DIN
AT
E G
EO
ME
TR
Y G
ra
ph
ea
ch
qu
ad
ril
ate
ra
l w
ith
th
e g
ive
n v
erti
ce
s.
De
term
ine
wh
eth
er t
he
fig
ure
is a
pa
ra
lle
log
ra
m.
Ju
sti
fy y
ou
r a
nsw
er w
ith
th
e
me
tho
d i
nd
ica
ted
.
5. P
(0,
0),
Q(3
, 4),
S(7
, 4),
Y(4
, 0);
Slo
pe F
orm
ula
6. S
(-2,
1),
R(1
, 3),
T(2
, 0),
Z(-
1,
-2);
Dis
tan
ce a
nd
Slo
pe F
orm
ula
s
7. W
(2,
5),
R(3
, 3),
Y(-
2,
-3),
N(-
3,
1);
Mid
poin
t F
orm
ula
ALG
EB
RA
F
ind
x a
nd
y s
o t
ha
t e
ach
qu
ad
ril
ate
ra
l is
a p
ara
lle
log
ra
m.
8.
9.
10
.
11
.
x+
16
y+
19
2y
2x
-8
2y-
11
y+
3
4x-
3
3x
( 4x
- 3
5) °
( 3x
+ 1
0) °
( 2y
- 5
) °
( y+
15) °
3x
- 1
4
x+
20
3y
+ 2
y+
20
6 -3
Y
es;
a p
air
of
op
po
sit
e s
ides
Yes;
bo
th p
air
s o
f o
pp
osit
e
is p
ara
llel
an
d c
on
gru
en
t.
an
gle
s a
re c
on
gru
en
t.
N
o;
no
ne o
f th
e t
ests
fo
r
Y
es;
bo
th p
air
s o
f o
pp
osit
e s
ides
para
llelo
gra
ms i
s f
ulf
ille
d.
are
co
ng
ruen
t.
Y
es;
SR
= Z
T a
nd
th
e s
lop
es o
f −
−
SR
an
d −
−
ZT a
re e
qu
al, s
o o
ne p
air
of
op
po
sit
e s
ides i
s p
ara
llel
an
d c
og
ruen
t.
x
= 2
4,
y =
19
x =
3,
y =
14
x
= 4
5,
y =
20
x =
17,
y =
9
N
o;
the m
idp
oin
ts o
f th
e d
iag
on
als
are
no
t th
e s
am
e p
oin
t.
Y
es;
the s
lop
e o
f −
−
PY
an
d −
−
QS
are
eq
ual
an
d t
he s
lop
e o
f −
−
PQ
an
d −
−
YS
are
eq
ual, s
o −
−
PY
‖ −
−
QS
an
d −
−
PQ
‖ −
−
YS
. O
pp
osit
e s
ides a
re p
ara
llel.
S
ee s
tud
en
ts’
gra
ph
s.
Answers (Lesson 6-3)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Chapter 6 A9 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
20
G
len
co
e G
eo
me
try
De
term
ine
wh
eth
er e
ach
qu
ad
ril
ate
ra
l is
a p
ara
lle
log
ra
m.
Ju
sti
fy y
ou
r a
nsw
er.
1.
2.
3
.
4.
CO
OR
DIN
AT
E G
EO
ME
TR
Y G
ra
ph
ea
ch
qu
ad
ril
ate
ra
l w
ith
th
e g
ive
n v
erti
ce
s.
De
term
ine
wh
eth
er t
he
fig
ure
is a
pa
ra
lle
log
ra
m.
Ju
sti
fy y
ou
r a
nsw
er w
ith
th
e
me
tho
d i
nd
ica
ted
.
5. P
(-5,
1),
S(-
2,
2),
F(-
1, -
3),
T(2
, -
2);
Slo
pe F
orm
ula
6
. R
(-2,
5),
O(1
, 3),
M(-
3, -
4),
Y(-
6, -
2);
Dis
tan
ce a
nd
Slo
pe F
orm
ula
s
ALG
EB
RA
F
ind
x a
nd
y s
o t
ha
t th
e q
ua
dril
ate
ra
l is
a p
ara
lle
log
ra
m.
7.
8.
9
.
10.
11
. T
ILE
DE
SIG
N T
he p
att
ern
sh
ow
n i
n t
he f
igu
re i
s to
con
sist
of
con
gru
en
t p
ara
llelo
gra
ms.
How
can
th
e d
esi
gn
er
be c
ert
ain
th
at
the s
hap
es
are
p
ara
llelo
gra
ms?
11
8°
62
°
11
8°
62
°
( 5x
+ 2
9) °
( 7x
- 1
1) °
( 3y
+ 1
5) °
( 5y
- 9
) °
3y-
5
-3x
+ 4
-4x
- 2
2y+
8
-4
x+
6
-6
x
7y
+ 3
12
y-
7-
4y-
2
x+
12
-2x
+ 6
y+
23
Practi
ce
Tests
fo
r P
ara
llelo
gra
ms
6 -3
Y
es;
the d
iag
on
als
bis
ect
No
; n
on
e o
f th
e t
ests
fo
r
each
oth
er.
para
llelo
gra
ms i
s f
ulf
ille
d.
Y
es;
bo
th p
air
s o
f o
pp
osit
e
No
; n
on
e o
f th
e t
ests
fo
r
an
gle
s a
re c
on
gru
en
t.
p
ara
llelo
gra
ms i
s f
ulf
ille
d.
y
es
x
= 2
0,
y =
12
x =
-6,
y =
13
x
= -
3,
y =
2
x =
-2,
y =
-5
S
am
ple
an
sw
er:
Co
nfi
rm t
hat
bo
th p
air
s o
f o
pp
osit
e !
are
#.
y
es
S
ee s
tud
en
ts’
wo
rk
Lesson 6-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
21
G
len
co
e G
eo
me
try
1
. B
ALA
NC
ING
N
ikia
, M
ad
ison
, A
ngela
, an
d S
helb
y a
re b
ala
nci
ng t
hem
selv
es
on
an
“X
”-sh
ap
ed
flo
ati
ng o
bje
ct.
To
bala
nce
th
em
selv
es,
th
ey w
an
t to
m
ak
e t
hem
selv
es
the v
ert
ices
of
a
para
llelo
gra
m.
In o
rder
to a
chie
ve t
his
, d
o a
ll f
ou
r of
them
have t
o b
e t
he s
am
e d
ista
nce
fro
m
the c
en
ter
of
the o
bje
ct?
Exp
lain
.
2
. C
OM
PA
SS
ES
T
wo c
om
pass
need
les
pla
ced
sid
e b
y s
ide o
n a
table
are
both
2 i
nch
es
lon
g a
nd
poin
t d
ue n
ort
h.
Do
they f
orm
th
e s
ides
of
a p
ara
llelo
gra
m?
3
. FO
RM
AT
ION
F
ou
r je
ts a
re f
lyin
g i
n
form
ati
on
. T
hre
e o
f th
e j
ets
are
sh
ow
n
in t
he g
rap
h.
If t
he f
ou
r je
ts a
re l
oca
ted
at
the v
ert
ices
of
a p
ara
llelo
gra
m,
wh
at
are
th
e t
hre
e p
oss
ible
loca
tion
s of
the
mis
sin
g j
et?
4
. S
TR
EE
T L
AM
PS
W
hen
a c
oord
inate
p
lan
e i
s p
lace
d o
ver
the H
arr
isvil
le t
ow
n
map
, th
e f
ou
r st
reet
lam
ps
in t
he c
en
ter
are
loca
ted
as
show
n.
Do t
he f
ou
r la
mp
s fo
rm t
he v
ert
ices
of
a p
ara
llelo
gra
m?
Exp
lain
.
5
. P
ICT
UR
E F
RA
ME
Aaro
n i
s m
ak
ing a
w
ood
en
pic
ture
fra
me i
n t
he s
hap
e o
f a
para
llelo
gra
m.
He h
as
two p
iece
s of
wood
th
at
are
3 f
eet
lon
g a
nd
tw
o t
hat
are
4 f
eet
lon
g.
a.
If h
e c
on
nect
s th
e p
iece
s of
wood
at
their
en
ds
to e
ach
oth
er,
in
wh
at
ord
er
mu
st h
e c
on
nect
th
em
to m
ak
e
a p
ara
llelo
gra
m?
b.
How
man
y d
iffe
ren
t p
ara
llelo
gra
ms
cou
ld h
e m
ak
e w
ith
th
ese
fou
r le
ngth
s of
wood
?
c.
Exp
lain
som
eth
ing A
aro
n m
igh
t d
o
to s
peci
fy p
reci
sely
th
e s
hap
e o
f th
e
para
llelo
gra
m.
5
5y
x
O–
5
–5
5
5
y
xO
Shel
by
Mad
iso
n
Nik
ia
Angel
a
6 -3
Wo
rd
Pro
ble
m P
racti
ce
Tests
fo
r P
ara
llelo
gra
ms
N
o,
Mad
iso
n a
nd
An
gela
have t
o
be t
he s
am
e d
ista
nce a
nd
Nik
ia
an
d S
helb
y h
ave t
o b
e t
he s
am
e
dis
tan
ce b
ut
Nik
ia a
nd
Sh
elb
y’s
dis
tan
ce f
rom
th
e c
en
ter
do
es
no
t h
ave t
o b
e e
qu
al
to M
ad
iso
n
an
d A
ng
ela
’s d
ista
nce.
y
es
(
1,
7),
(9,
-1),
(-
7,
-3)
Y
es;
the l
en
gth
s o
f th
e o
pp
osit
e
sid
es a
re c
on
gru
en
t.
He m
ust
alt
ern
ate
th
e l
en
gth
s
3,
4,
3,
4 o
r 4,
3,
4,
3.
infi
nit
ely
man
y
Sam
ple
an
sw
er:
He c
ou
ld
sp
ecif
y t
he l
en
gth
of
a
dia
go
nal.
Answers (Lesson 6-3)
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Chapter 6 A10 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
22
G
len
co
e G
eo
me
try
En
rich
men
t
Tests
fo
r P
ara
llelo
gra
ms
By d
efi
nit
ion
, a q
uad
rila
tera
l is
a p
ara
llelo
gra
m i
f a
nd
on
ly i
f both
pair
s of
op
posi
te s
ides
are
para
llel.
Wh
at
con
dit
ion
s oth
er
than
both
pair
s of
op
posi
te s
ides
para
llel
wil
l gu
ara
nte
e
that
a q
uad
rila
tera
l is
a p
ara
llelo
gra
m?
In t
his
act
ivit
y,
severa
l p
oss
ibil
itie
s w
ill
be
invest
igate
d b
y d
raw
ing q
uad
rila
tera
ls t
o s
ati
sfy c
ert
ain
con
dit
ion
s. R
em
em
ber
that
an
y
test
th
at
seem
s to
work
is
not
gu
ara
nte
ed
to w
ork
un
less
it
can
be f
orm
all
y p
roven
.
Co
mp
lete
.
1. D
raw
a q
uad
rila
tera
l w
ith
on
e p
air
of
op
posi
te s
ides
con
gru
en
t.
Mu
st i
t be a
para
llelo
gra
m?
2. D
raw
a q
uad
rila
tera
l w
ith
both
pair
s of
op
posi
te s
ides
con
gru
en
t.
Mu
st i
t be a
para
llelo
gra
m?
3. D
raw
a q
uad
rila
tera
l w
ith
on
e p
air
of
op
posi
te s
ides
para
llel
an
d
the o
ther
pair
of
op
posi
te s
ides
con
gru
en
t. M
ust
it
be a
p
ara
llelo
gra
m?
4. D
raw
a q
uad
rila
tera
l w
ith
on
e p
air
of
op
posi
te s
ides
both
para
llel
an
d c
on
gru
en
t. M
ust
it
be a
para
llelo
gra
m?
5. D
raw
a q
uad
rila
tera
l w
ith
on
e p
air
of
op
posi
te a
ngle
s co
ngru
en
t.
Mu
st i
t be a
para
llelo
gra
m?
6. D
raw
a q
uad
rila
tera
l w
ith
both
pair
s of
op
posi
te a
ngle
s co
ngru
en
t.
Mu
st i
t be a
para
llelo
gra
m?
7. D
raw
a q
uad
rila
tera
l w
ith
on
e p
air
of
op
posi
te s
ides
para
llel
an
d
on
e p
air
of
op
posi
te a
ngle
s co
ngru
en
t. M
ust
it
be a
para
llelo
gra
m?
6 -3
no
yes
no
yes
no
yes
yes
Lesson 6-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
23
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Recta
ng
les
Pro
pe
rtie
s o
f R
ect
an
gle
s A
re
cta
ng
le i
s a q
uad
rila
tera
l w
ith
fou
r
righ
t an
gle
s. H
ere
are
th
e p
rop
ert
ies
of
rect
an
gle
s.
A r
ect
an
gle
has
all
th
e p
rop
ert
ies
of
a p
ara
llelo
gra
m.
• O
pp
osi
te s
ides
are
para
llel.
• O
pp
osi
te a
ngle
s are
con
gru
en
t.
• O
pp
osi
te s
ides
are
con
gru
en
t.
• C
on
secu
tive a
ngle
s are
su
pp
lem
en
tary
.
• T
he d
iagon
als
bis
ect
each
oth
er.
Als
o:
• A
ll f
ou
r an
gle
s are
rig
ht
an
gle
s.
∠U
TS
, ∠
TS
R,
∠S
RU
, an
d ∠
RU
T a
re r
igh
t an
gle
s.
• T
he d
iagon
als
are
con
gru
en
t.
−−
TR
# −
−−
US
TS R
U
Q
Q
ua
dril
ate
ra
l R
UT
S
ab
ov
e i
s a
re
cta
ng
le.
If U
S =
6x +
3 a
nd
RT
= 7
x -
2,
fin
d x
.
Th
e d
iagon
als
of
a r
ect
an
gle
are
con
gru
en
t,
so U
S =
RT
.
6x +
3 =
7x -
2
3 =
x -
2
5 =
x
Q
ua
dril
ate
ra
l R
UT
S
ab
ov
e i
s a
re
cta
ng
le.
If m∠
ST
R =
8x +
3
an
d m∠
UT
R =
16x
- 9
, fi
nd
m∠
ST
R.
∠U
TS
is
a r
igh
t an
gle
, so
m
∠S
TR
+ m
∠U
TR
= 9
0.
8x +
3 +
16
x -
9 =
90
24
x -
6 =
90
24
x =
96
x =
4
m∠
ST
R =
8x +
3 =
8(4
) +
3 o
r 35
Exerc
ises
Qu
ad
ril
ate
ra
l A
BC
D i
s a
re
cta
ng
le.
1. If
AE
= 3
6 a
nd
CE
= 2
x -
4,
fin
d x
.
2. If
BE
= 6
y +
2 a
nd
CE
= 4
y +
6,
fin
d y
.
3. If
BC
= 2
4 a
nd
AD
= 5
y -
1,
fin
d y
.
4. If
m∠
BE
A =
62,
fin
d m
∠B
AC
.
5. If
m∠
AE
D =
12
x a
nd
m∠
BE
C =
10
x +
20,
fin
d m
∠A
ED
.
6. If
BD
= 8
y -
4 a
nd
AC
= 7
y +
3,
fin
d B
D.
7. If
m∠
DB
C =
10
x a
nd
m∠
AC
B =
4x
2-
6,
fin
d m
∠ A
CB
.
8. If
AB
= 6
y a
nd
BC
= 8
y,
fin
d B
D i
n t
erm
s of
y.
BC D
A
E
6 -4
Exam
ple
1Exam
ple
2
20
2
5
59
120
52
30
10
y
Answers (Lesson 6-3 and Lesson 6-4)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Chapter 6 A11 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
24
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Recta
ng
les
Pro
ve
th
at
Pa
rall
elo
gra
ms
Are
Re
cta
ng
les
Th
e d
iagon
als
of
a r
ect
an
gle
are
co
ngru
en
t, a
nd
th
e c
on
vers
e i
s als
o t
rue.
If t
he
dia
go
na
ls o
f a
pa
ralle
log
ram
are
co
ng
rue
nt,
th
en
th
e p
ara
llelo
gra
m is a
re
cta
ng
le.
In t
he c
oord
inate
pla
ne y
ou
can
use
th
e D
ista
nce
Form
ula
, th
e S
lop
e F
orm
ula
, an
d
pro
pert
ies
of
dia
gon
als
to s
how
th
at
a f
igu
re i
s a r
ect
an
gle
.
Q
ua
dril
ate
ra
l A
BC
D h
as v
erti
ce
s A
(-3
, 0
), B
(-2
, 3
),
C(4
, 1
), a
nd
D(3
, -
2).
De
term
ine
wh
eth
er A
BC
D i
s a
re
cta
ng
le.
Meth
od
1:
Use
th
e S
lop
e F
orm
ula
.
slop
e o
f −
−
AB
=
3 -
0
−
-2 -
(-
3)
= 3
−
1 o
r 3
slop
e o
f −−
− A
D =
-
2 -
0
−
3 -
(-
3)
= -
2
−
6 o
r -
1
−
3
slop
e o
f −−
− C
D =
-
2 -
1
−
3 -
4
= -
3
−
-1 o
r 3
slop
e o
f −−
− B
C =
1 -
3
−
4 -
(-
2)
= -
2
−
6 o
r -
1
−
3
Op
posi
te s
ides
are
para
llel,
so t
he f
igu
re i
s a p
ara
llelo
gra
m.
Con
secu
tive s
ides
are
p
erp
en
dic
ula
r, s
o A
BC
D i
s a r
ect
an
gle
.
x
y O
D
B
A
EC
6 -4 Exam
ple
Me
tho
d 2
: U
se t
he D
ista
nce
Form
ula
.
AB
= √
##
##
##
##
##
(-
3 -
(-
2) )
2 +
(0 -
3 ) 2
or
√ #
#
10
BC
= √
##
##
##
##
(-
2 -
4 ) 2
+ (
3 -
1 ) 2
or
√ #
#
40
CD
= √
##
##
##
##
#
(4
- 3
) 2 +
(1 -
(-
2) )
2
or
√ #
#
10
AD
= √
##
##
##
##
##
(-3 -
3 ) 2
+ (
0 -
(-
2) )
2
or
√ #
#
40
Op
posi
te s
ides
are
con
gru
en
t, t
hu
s A
BC
D i
s a p
ara
llelo
gra
m.
AC
= √
##
##
##
##
(-
3 -
4 ) 2
+ (
0 -
1 ) 2
or
√ #
#
50
BD
= √
##
##
##
##
##
(-2 -
3 ) 2
+ (
3 -
(-
2) )
2
or
√ #
#
50
AB
CD
is
a p
ara
llelo
gra
m w
ith
con
gru
en
t d
iagon
als
, so
AB
CD
is
a r
ect
an
gle
.
Exerc
ises
CO
OR
DIN
AT
E G
EO
ME
TR
Y G
ra
ph
ea
ch
qu
ad
ril
ate
ra
l w
ith
th
e g
ive
n v
erti
ce
s.
De
term
ine
wh
eth
er t
he
fig
ure
is a
re
cta
ng
le.
Ju
sti
fy y
ou
r a
nsw
er u
sin
g t
he
ind
ica
ted
fo
rm
ula
.
1
. A
(−3,
1),
B(−
3,
3),
C(3
, 3),
D(3
, 1);
Dis
tan
ce F
orm
ula
2
. A
(−3,
0),
B(−
2,
3),
C(4
, 5),
D(3
, 2);
Slo
pe F
orm
ula
3
. A
(−3,
0),
B(−
2,
2),
C(3
, 0),
D(2
−2);
Dis
tan
ce F
orm
ula
4. A
(−1,
0),
B(0
, 2),
C(4
, 0),
D(3
, −
2);
Dis
tan
ce F
orm
ula
Y
es;
AB
= 2
, B
C =
6,
CD
= 2
, D
A =
6,
AC
= √
""
40 ,
BD
= √
""
40 ,
op
po
sit
e
sid
es a
nd
dia
go
nals
are
co
ng
ruen
t.
N
o;
slo
pe o
f −
−
AB
= 3
, slo
pe o
f −
−
BC
= 1
−
3 ,
slo
pes s
ho
w t
hat
two
co
nsecu
tive
sid
es a
re n
ot
perp
en
dic
ula
r.
S
ee s
tud
en
ts’
wo
rk
N
o;
AC
= 6
, B
D =
√ "
"
32 ,
dia
go
nals
are
no
t co
ng
ruen
t.
CD
= √
"
5 ,
DA
= √
""
20 ,
AC
= 5
, B
D =
5,
op
po
sit
e s
ides a
nd
dia
go
nals
are
co
ng
ruen
t.
Y
es;
AB
= √
"
5 ,
BC
= √
""
20 ,
Lesson 6-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
25
G
len
co
e G
eo
me
try
Sk
ills
Pra
ctic
e
Recta
ng
les
ALG
EB
RA
Q
ua
dril
ate
ra
l A
BC
D i
s a
re
cta
ng
le.
1. If
AC
= 2
x +
13 a
nd
DB
= 4
x -
1,
fin
d D
B.
2. If
AC
= x
+ 3
an
d D
B =
3x -
19,
fin
d A
C.
3. If
AE
= 3
x +
3 a
nd
EC
= 5
x -
15,
fin
d A
C.
4. If
DE
= 6
x -
7 a
nd
AE
= 4
x +
9,
fin
d D
B.
5. If
m∠
DA
C =
2x +
4 a
nd
m∠
BA
C =
3x +
1,
fin
d m
∠B
AC
.
6. If
m∠
BD
C =
7x +
1 a
nd
m∠
AD
B =
9x -
7,
fin
d m
∠B
DC
.
7. If
m∠
AB
D =
7x -
31 a
nd
m∠
CD
B =
4x +
5,
fin
d m
∠A
BD
.
8. If
m∠
BA
C =
x +
3 a
nd
m∠
CA
D =
x +
15,
fin
d m
∠B
AC
.
9
. PR
OO
F:
Wri
te a
tw
o-c
olu
mn
pro
of.
G
ive
n:
RS
TV
is
a r
ect
an
gle
an
d U
is
the
mid
poin
t of
−−
VT
.
P
ro
ve
: △
RU
V '
△S
UT
P
roo
f
S
tate
me
nts
R
ea
so
ns
1
. R
STV
is a
recta
ng
le.
1.
Giv
en
2
. ∠
V a
nd
∠T a
re r
igh
t an
gle
s.
2.
Defi
nit
ion
of
recta
ng
le
3.
∠V
% ∠
T
3.
All r
t &
are
%.
4.
U i
s t
he m
idp
oin
t o
f −
−
VT .
4.
Giv
en
5
. −
−
VU
% −
−
TU
5.
Defi
nit
ion
of
mid
po
int
6.
−−
VR
% −
−
TS
6.
Op
p s
ides o
f '
are
co
ng
ruen
t.
7
. △
RU
V %
△S
UT
7.
SA
S
CO
OR
DIN
AT
E G
EO
ME
TR
Y G
ra
ph
ea
ch
qu
ad
ril
ate
ra
l w
ith
th
e g
ive
n v
erti
ce
s.
De
term
ine
wh
eth
er t
he
fig
ure
is a
re
cta
ng
le.
Ju
sti
fy y
ou
r a
nsw
er u
sin
g t
he
in
dic
ate
d f
orm
ula
.
10. P
(-3,
-2),
Q(-
4,
2),
R(2
, 4),
S(3
, 0);
Slo
pe F
orm
ula
11
. J
(-6,
3),
K(0
, 6),
L(2
, 2),
M(-
4,
-1);
Dis
tan
ce F
orm
ula
12
. T
(4,
1),
U(3
, -
1),
X(-
3,
2),
Y(-
2,
4);
Dis
tan
ce F
orm
ula
DCB
AE
6-4
27
14 60
82
52
43 5
3
39
N
o;
Sam
ple
an
sw
er:
An
gle
s a
re n
ot
rig
ht
an
gle
s.
Y
es;
Sam
ple
an
sw
er:
Bo
th p
air
s o
f o
pp
osit
e s
ides a
re c
on
gru
en
t
an
d d
iag
on
als
are
co
ng
ruen
t.
Y
es;
Sam
ple
an
sw
er:
Bo
th p
air
s o
f o
pp
osit
e s
ides a
re c
on
gru
en
t an
d t
he
dia
go
nals
are
co
ng
ruen
t.
S
ee s
tud
en
ts’
gra
ph
s.
Answers (Lesson 6-4)
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Chapter 6 A12 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
26
G
len
co
e G
eo
me
try
Practi
ce
Recta
ng
les
ALG
EB
RA
Q
ua
dril
ate
ra
l RSTU
is a
re
cta
ng
le.
1. If
UZ=x+
21 a
nd
ZS=
3x-
15,
fin
d US
.
2. If
RZ
= 3x +
8 a
nd
ZS
= 6x -
28,
fin
d UZ
.
3. If
RT
= 5x +
8 a
nd
RZ
= 4x +
1,
fin
d ZT
.
4. If
m∠SUT
= 3x +
6 a
nd
m∠RUS
= 5x -
4,
fin
d m∠SUT
.
5. If
m∠SRT
= x
+ 9
an
d m∠UTR
= 2x -
44,
fin
d m∠UTR
.
6. If
m∠RSU
= x
+ 4
1 a
nd
m∠TUS
= 3x +
9,
fin
d m∠RSU
.
Qu
ad
ril
ate
ra
l GHJK
is a
re
cta
ng
le.
Fin
d e
ach
me
asu
re
if m∠
1 =
37
.
7. m∠
2
8. m∠
3
9. m∠
4
10. m∠
5
11
. m∠
6
12. m∠
7
CO
OR
DIN
AT
E G
EO
ME
TR
Y G
ra
ph
ea
ch
qu
ad
ril
ate
ra
l w
ith
th
e g
ive
n v
erti
ce
s.
De
term
ine
wh
eth
er t
he
fig
ure
is a
re
cta
ng
le.
Ju
sti
fy y
ou
r a
nsw
er u
sin
g t
he
ind
ica
ted
fo
rm
ula
.
13. B
(-4,
3),
G(-
2,
4),
H(1
, -
2),
L(-
1, -
3);
Slo
pe F
orm
ula
14
. N
(-4,
5),
O(6
, 0),
P(3
, -
6),
Q(-
7, -
1);
Dis
tan
ce F
orm
ula
15
. C
(0,
5),
D(4
, 7),
E(5
, 4),
F(1
, 2);
Slo
pe F
orm
ula
16. LA
ND
SC
AP
ING
H
un
tin
gto
n P
ark
off
icia
ls a
pp
roved
a r
ect
an
gu
lar
plo
t of
lan
d f
or
a
Jap
an
ese
Zen
gard
en
. Is
it
suff
icie
nt
to k
now
th
at
op
posi
te s
ides
of
the g
ard
en
plo
t are
co
ngru
en
t an
d p
ara
llel
to d
ete
rmin
e t
hat
the g
ard
en
plo
t is
rect
an
gu
lar?
Exp
lain
.
UTS
RZ
K
21
5
43
67
JHG
6 -4
78
44
9
39
62
57
53
37
37
53
106
74
Y
es;
sam
ple
an
sw
er:
Op
po
sit
e s
ides a
re p
ara
llel
an
d c
on
secu
tive s
ides
are
perp
en
dic
ula
r.
Y
es;
sam
ple
an
sw
er:
Op
po
sit
e s
ides a
re c
on
gru
en
t an
d d
iag
on
als
are
co
ng
ruen
t.
N
o;
sam
ple
an
sw
er:
Dia
go
nals
are
no
t co
ng
ruen
t.
N
o;
if y
ou
on
ly k
no
w t
hat
op
po
sit
e s
ides a
re c
on
gru
en
t an
d p
ara
llel, t
he
mo
st
yo
u c
an
co
nclu
de i
s t
hat
the p
lot
is a
para
llelo
gra
m.
S
ee s
tud
en
ts’
wo
rk
Lesson 6-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
27
G
len
co
e G
eo
me
try
1
. FR
AM
ES
Jale
n m
ak
es
the r
ect
an
gu
lar
fram
e s
how
n.
In o
rder
to m
ak
e s
ure
th
at
it i
s a
rect
an
gle
, Jale
n m
easu
res
the d
ista
nce
s BD
an
d AC
. H
ow
sh
ou
ld t
hese
tw
o
dis
tan
ces
com
pare
if
the f
ram
e i
s a
rect
an
gle
?
2
. B
OO
KS
HE
LV
ES
A
book
shelf
con
sist
s of
two v
ert
ical
pla
nk
s w
ith
fiv
e h
ori
zon
tal
shelv
es.
Are
each
of
the
fou
r se
ctio
ns
for
book
s re
ctan
gle
s? E
xp
lain
.
3
. LA
ND
SC
AP
ING
A
lan
dsc
ap
er
is
mark
ing o
ff t
he c
orn
ers
of
a r
ect
an
gu
lar
plo
t of
lan
d.
Th
ree o
f th
e c
orn
ers
are
in
p
lace
as
show
n.
Wh
at
are
th
e c
oord
inate
s of
the f
ou
rth
co
rner?
4
. S
WIM
MIN
G P
OO
LS
A
nto
nio
is
desi
gn
ing a
sw
imm
ing p
ool
on
a
coord
inate
gri
d.
Is i
t a r
ect
an
gle
? E
xp
lain
.
5
. P
AT
TE
RN
SV
ero
nic
a m
ad
e t
he p
att
ern
sh
ow
n o
ut
of
7 r
ect
an
gle
s w
ith
fou
r equ
al
sid
es.
Th
e s
ide l
en
gth
of
each
re
ctan
gle
is
wri
tten
in
sid
e t
he r
ect
an
gle
.
a.
How
man
y r
ect
an
gle
s ca
n b
e f
orm
ed
u
sin
g t
he l
ines
in t
his
fig
ure
?
b.
If V
ero
nic
a w
an
ted
to e
xte
nd
her
patt
ern
by a
dd
ing a
noth
er
rect
an
gle
w
ith
4 e
qu
al
sid
es
to m
ak
e a
larg
er
rect
an
gle
, w
hat
are
th
e p
oss
ible
si
de l
en
gth
s of
rect
an
gle
s th
at
she
can
ad
d?
Wo
rd
Pro
ble
m P
racti
ce
Recta
ng
les
1 1
2 3
5
85y
xO
5
5y
xO
–5
–5
AB
DC
6 -4
T
hey s
ho
uld
be e
qu
al.
Y
es;
each
of
the f
ou
r an
gle
s o
f
each
recta
ng
le i
s c
reate
d b
y a
str
aig
ht
ho
rizo
nta
l lin
e a
nd
a
str
aig
ht
vert
ical
lin
e,
so
each
has f
ou
r 90
° an
gle
s.
(-6,
3)
Y
es.
Sam
ple
an
sw
er:
Th
e s
lop
e o
f
the l
on
g s
ides i
s 3
an
d t
he s
lop
e
o
f th
e s
ho
rt s
ides i
s -
1
−
3 ,
so
each
p
air
of
sid
es i
s p
ara
llel. A
lso
, th
e
lon
g a
nd
sh
ort
sid
es h
ave s
lop
es
that
are
perp
en
dic
ula
r to
each
oth
er,
makin
g t
he f
ou
r co
rners
rig
ht
an
gle
s.
11
8 a
nd
13 u
nit
s
Answers (Lesson 6-4)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Chapter 6 A13 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
28
G
len
co
e G
eo
me
try
Co
nsta
nt
Peri
mete
rD
ou
gla
s w
an
ts t
o f
en
ce a
rect
an
gu
lar
regio
n o
f h
is b
ack
yard
for
his
dog.
He b
ou
gh
t 200 f
eet
of
fen
ce.
1. C
om
ple
te t
he t
able
to s
how
th
e
dim
en
sion
s of
five d
iffe
ren
t re
ctan
gu
lar
pen
s th
at
wou
ld u
se t
he e
nti
re 2
00 f
eet
of
fen
ce.
Th
en
fin
d t
he a
rea o
f each
re
ctan
gu
lar
pen
.
2. D
o a
ll f
ive o
f th
e r
ect
an
gu
lar
pen
s h
ave
the s
am
e a
rea?
If n
ot,
wh
ich
on
e h
as
the l
arg
er
are
a?
3. W
rite
a r
ule
for
fin
din
g t
he d
imen
sion
s of
a r
ect
an
gle
wit
h
the l
arg
est
poss
ible
are
a f
or
a g
iven
peri
mete
r.
4. L
et x r
ep
rese
nt
the l
en
gth
of
a r
ect
an
gle
an
d y
th
e w
idth
. W
rite
th
e f
orm
ula
for
all
rect
an
gle
s w
ith
a p
eri
mete
r of
200.
Th
en
gra
ph
th
is r
ela
tion
ship
on
th
e c
oord
inate
pla
ne
at
the r
igh
t.
Ju
lio r
ead
th
at
a d
og t
he s
ize o
f h
is n
ew
pet,
Ben
nie
, sh
ou
ld h
ave a
t le
ast
100 s
qu
are
feet
in
his
pen
. B
efo
re g
oin
g t
o t
he s
tore
to b
uy f
en
ce,
Ju
lio m
ad
e a
table
to d
ete
rmin
e t
he
dim
en
sion
s fo
r B
en
nie
’s r
ect
an
gu
lar
pen
.
5. C
om
ple
te t
he t
able
to f
ind
fiv
e p
oss
ible
d
imen
sion
s of
a r
ect
an
gu
lar
fen
ced
are
a o
f 100 s
qu
are
feet.
6. Ju
lio w
an
ts t
o s
ave m
on
ey b
y p
urc
hasi
ng
the l
east
nu
mber
feet
of
fen
cin
g t
o e
ncl
ose
th
e
100 s
qu
are
feet.
Wh
at
wil
l be t
he d
imen
sion
s of
the c
om
ple
ted
pen
?
7. W
rite
a r
ule
for
fin
din
g t
he d
imen
sion
s of
a r
ect
an
gle
wit
h
the l
east
poss
ible
peri
mete
r fo
r a g
iven
are
a.
8. F
or
len
gth
x a
nd
wid
th y
, w
rite
a f
orm
ula
for
the a
rea o
f a
rect
an
gle
wit
h a
n a
rea o
f 100 s
qu
are
feet.
Th
en
gra
ph
th
e
form
ula
.
100
100
y
xOy
xO
100
100
6-4
Pe
rim
ete
rL
en
gth
Wid
thA
rea
20
08
0
20
07
0
20
06
0
20
05
0
20
04
5
Are
aL
en
gth
Wid
thH
ow
mu
ch
fen
ce
to
bu
y
10
0
10
0
10
0
10
0
10
0
En
rich
men
t
N
o,
the 5
0 ×
50 p
en
has t
he l
arg
est
are
a.
T
he r
ecta
ng
le w
ith
th
e l
arg
est
are
a f
or
a g
iven
peri
mete
r is
a s
qu
are
.
2
x +
2y =
200 o
r x +
y =
100
S
ee t
ab
le f
or
sam
ple
an
sw
ers
.
1
0 f
t ×
10 f
t
T
he r
ecta
ng
le w
ith
th
e l
east
po
ssib
le p
eri
mete
r fo
r
a g
iven
are
a i
s a
sq
uare
.
x
y=
10
0
11
00
202
250
104
425
58
520
50
10
10
40
20
1600
30
2100
40
2400
50
2500
55
2475
Lesson 6-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
29
G
len
co
e G
eo
me
try
Gra
ph
ing C
alc
ula
tor
Act
ivit
y
TI-
Nsp
ire:
Exp
lori
ng
Recta
ng
les
A q
uad
rila
tera
l w
ith
fou
r ri
gh
t an
gle
s is
a r
ecta
ng
le.
Th
e T
I-N
spir
e c
an
be u
sed
to e
xp
lore
so
me o
f th
e c
hara
cteri
stic
s of
a r
ect
an
gle
. U
se t
he f
oll
ow
ing s
tep
s to
dra
w a
rect
an
gle
.
Ste
p 1
S
et
up
th
e c
alc
ula
tor
in t
he c
orr
ect
mod
e.
• C
hoose
Gra
ph
s &
Ge
om
etr
y f
rom
th
e H
om
e M
en
u.
• F
rom
th
e V
iew
men
u, ch
oose
4:
Hid
e A
xis
Ste
p 2
D
raw
th
e r
ect
an
gle
• F
rom
th
e 8
: S
ha
pe
s m
en
u c
hoose
3:
Re
cta
ng
le.
• C
lick
on
ce t
o d
efi
ne t
he c
orn
er
of
the r
ect
an
gle
. T
hen
move a
nd
cli
ck a
gain
. T
he
sid
e o
f th
e r
ect
an
gle
is
now
defi
ned
. M
ove p
erp
en
dic
ula
rly t
o d
raw
th
e r
ect
an
gle
. C
lick
to a
nch
or
the s
hap
e.
Ste
p 3
M
easu
re t
he l
en
gth
s of
the s
ides
of
the r
ect
an
gle
.
• F
rom
th
e 7
: M
ea
su
re
me
nt
men
u c
hoose
1:
Le
ng
th (
Note
th
at
wh
en
you
scr
oll
over
the r
ect
an
gle
, th
e v
alu
e n
ow
sh
ow
n i
s th
e p
eri
mete
r of
the r
ect
an
gle
.)
• S
ele
ct e
ach
en
dp
oin
t of
a s
egm
en
t of
the r
ect
an
gle
. T
hen
cli
ck o
r p
ress
En
ter
to
an
chor
the l
en
gth
of
the s
egm
en
t in
th
e w
ork
are
a.
• R
ep
eat
for
the o
ther
sid
es
of
the r
ect
an
gle
.
Exercis
es
1
. W
hat
ap
pears
to b
e t
rue a
bou
t th
e o
pp
osi
te s
ides
of
the r
ect
an
gle
?
2
. D
raw
th
e d
iagon
als
of
the r
ect
an
gle
usi
ng 5
: S
eg
me
nt
from
th
e 6
: P
oin
ts a
nd
Lin
es
Men
u.
Cli
ck o
n t
wo o
pp
osi
te v
ert
ices
to d
raw
th
e d
iagon
al.
Rep
eat
to d
raw
th
e o
ther
dia
gon
al.
a.
Measu
re e
ach
dia
gon
al
usi
ng t
he m
easu
rem
en
t to
ol.
Wh
at
do y
ou
obse
rve?
b.
Wh
at
is t
rue a
bou
t th
e t
rian
gle
s fo
rmed
by t
he s
ides
of
the r
ect
an
gle
an
d a
dia
gon
al?
Ju
stif
y y
ou
r co
ncl
usi
on
.
3
. P
ress
Cle
ar t
hre
e t
imes
an
d s
ele
ct Y
es t
o c
lear
the s
creen
. R
ep
eat
the s
tep
s an
d d
raw
an
oth
er
rect
an
gle
. D
o t
he r
ela
tion
ship
s th
at
you
fou
nd
for
the f
irst
rect
an
gle
you
dre
w
hold
tru
e f
or
this
rect
an
gle
?
6-4
T
he o
pp
osit
e s
ides o
f a r
ecta
ng
le a
re c
on
gru
en
t.
b
. T
he t
rian
gle
s a
re c
on
gru
en
t b
y S
SS
.
Y
es,
the o
pp
osit
e s
ides a
re c
on
gru
en
t an
d t
he d
iag
on
als
are
co
ng
ruen
t.
a
. T
he d
iag
on
als
of
a r
ecta
ng
le a
re c
on
gru
en
t.
Answers (Lesson 6-4)
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Chapter 6 A14 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
30
G
len
co
e G
eo
me
try
Geo
mete
r’s
Sk
etc
hp
ad
Act
ivit
y
Exp
lori
ng
Recta
ng
les
A q
uad
rila
tera
l w
ith
fou
r ri
gh
t an
gle
s is
a r
ecta
ng
le.
Th
e G
eom
ete
r’s
Sk
etc
hp
ad
is
a u
sefu
l
tool
for
exp
lori
ng s
om
e o
f th
e c
hara
cteri
stic
s of
a r
ect
an
gle
. U
se t
he f
oll
ow
ing s
tep
s to
dra
w
a r
ect
an
gle
.
Ste
p 1
U
se t
he L
ine t
ool
to d
raw
a l
ine a
nyw
here
on
th
e s
creen
.
Ste
p 2
U
se t
he P
oin
t to
ol
to d
raw
a p
oin
t th
at
is
not
on
th
e l
ine.
To d
raw
a l
ine p
erp
en
dic
ula
r
to t
he f
irst
lin
e y
ou
dre
w,
sele
ct t
he f
irst
lin
e
an
d t
he p
oin
t. T
hen
ch
oose
Pe
rp
en
dic
ula
r
Lin
e f
rom
th
e C
on
str
uct
men
u.
Ste
p 3
U
se t
he P
oin
t to
ol
to d
raw
a p
oin
t th
at
is
not
on
eit
her
of
the l
ines
you
have d
raw
n.
Rep
eat
the p
roce
du
re i
n S
tep
2 t
o d
raw
lin
es
perp
en
dic
ula
r to
th
e t
wo l
ines
you
have d
raw
n.
A r
ect
an
gle
is
form
ed
by t
he s
egm
en
ts w
hose
en
dp
oin
ts a
re t
he p
oin
ts o
f in
ters
ect
ion
of
the
lin
es.
Exerc
ises
Use
th
e m
ea
su
rin
g c
ap
ab
ilit
ies o
f T
he
Ge
om
ete
r’s
Sk
etc
hp
ad
to
ex
plo
re
the
ch
ara
cte
ris
tics o
f a
re
cta
ng
le.
1
. W
hat
ap
pears
to b
e t
rue a
bou
t th
e o
pp
osi
te s
ides
of
the r
ect
an
gle
th
at
you
dre
w?
Mak
e a
con
ject
ure
an
d t
hen
measu
re e
ach
sid
e t
o c
heck
you
r co
nje
ctu
re.
2
. D
raw
th
e d
iagon
als
of
the r
ect
an
gle
by u
sin
g t
he S
ele
ctio
n A
rrow
tool
to c
hoose
tw
o
op
posi
te v
ert
ices.
Th
en
ch
oose
Se
gm
en
t fr
om
th
e C
on
str
uct
men
u t
o d
raw
th
e
dia
gon
al.
Rep
eat
to d
raw
th
e o
ther
dia
gon
al.
a.
Measu
re e
ach
dia
gon
al.
Wh
at
do y
ou
obse
rve?
b.
Wh
at
is t
rue a
bou
t th
e t
rian
gle
s fo
rmed
by t
he s
ides
of
the r
ect
an
gle
an
d a
dia
gon
al?
Ju
stif
y y
ou
r co
ncl
usi
on
.
3
. C
hoose
Ne
w S
ke
tch
fro
m t
he F
ile
men
u a
nd
foll
ow
ste
ps
1–3 t
o d
raw
an
oth
er
rect
an
gle
. D
o t
he r
ela
tion
ship
s you
fou
nd
for
the f
irst
rect
an
gle
you
dre
w h
old
tru
e f
or
this
rect
an
gle
als
o?
6 -4
T
he o
pp
osit
e s
ides o
f a r
ecta
ng
le a
re c
on
gru
en
t.
Th
e d
iag
on
als
of
a r
ecta
ng
le a
re c
on
gru
en
t.
Th
e t
rian
gle
s a
re c
on
gru
en
t b
y S
SS
.
Y
es,
the o
pp
osit
e s
ides a
re c
on
gru
en
t an
d t
he d
iag
on
als
are
co
ng
ruen
t.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 6-5
Ch
ap
ter
6
31
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Rh
om
bi
an
d S
qu
are
s
Pro
pe
rtie
s o
f R
ho
mb
i a
nd
Sq
ua
res
A r
ho
mb
us i
s a q
uad
rila
tera
l w
ith
fou
r co
ngru
en
t si
des.
Op
posi
te s
ides
are
con
gru
en
t, s
o a
rh
om
bu
s is
als
o a
para
llelo
gra
m a
nd
has
all
of
the p
rop
ert
ies
of
a p
ara
llelo
gra
m.
Rh
om
bi
als
o h
ave t
he f
oll
ow
ing p
rop
ert
ies.
A s
qu
are
is
a p
ara
llelo
gra
m w
ith
fou
r co
ngru
en
t si
des
an
d f
ou
r
con
gru
en
t an
gle
s. A
squ
are
is
both
a r
ect
an
gle
an
d a
rh
om
bu
s;
there
fore
, all
pro
pert
ies
of
para
llelo
gra
ms,
rect
an
gle
s, a
nd
rh
om
bi
ap
ply
to s
qu
are
s.
In
rh
om
bu
s ABCD
, m
∠BAC
= 3
2.
Fin
d t
he
me
asu
re
of
ea
ch
nu
mb
ere
d a
ng
le.
ABCD
is
a r
hom
bu
s, s
o t
he d
iagon
als
are
perp
en
dic
ula
r an
d △
ABE
is a
rig
ht
tria
ngle
. T
hu
s m
∠4 =
90 a
nd
m∠
1 =
90 -
32 o
r 58.
Th
e
dia
gon
als
in
a r
hom
bu
s bis
ect
th
e v
ert
ex a
ngle
s, s
o m
∠1 =
m∠
2.
Th
us,
m∠
2 =
58
.
A r
hom
bu
s is
a p
ara
llelo
gra
m,
so t
he o
pp
osi
te s
ides
are
para
llel.
∠BAC
an
d ∠
3 a
re
alt
ern
ate
in
teri
or
an
gle
s fo
r p
ara
llel
lin
es,
so m
∠3 =
32
.
Exerc
ises
Qu
ad
ril
ate
ra
l ABCD
is a
rh
om
bu
s.
Fin
d e
ach
va
lue
or m
ea
su
re
.
1
. If
m∠ABD
= 6
0,
fin
d m
∠BDC
. 2. If
AE
= 8
, fi
nd
AC
.
3
. If
AB
= 2
6 a
nd
BD
= 2
0,
fin
d AE
. 4. F
ind
m∠CEB
.
5
. If
m∠CBD
= 5
8,
fin
d m
∠ACB
. 6. If
AE
= 3x -
1 a
nd
AC
= 1
6,
fin
d x
.
7. If
m∠CDB
= 6y a
nd
m∠ACB
= 2y +
10,
fin
d y
.
8. If
AD
= 2x +
4 a
nd
CD
= 4x -
4,
fin
d x
.
MO
HR
B
CA
D
E
B
32°
12
3
4
C
AD
EB
6 -5
Th
e d
iag
on
als
are
pe
rpe
nd
icu
lar.
−−
−
MH
⊥ −
−−
RO
Ea
ch
dia
go
na
l b
ise
cts
a p
air o
f o
pp
osite
an
gle
s.
−−
−
MH
bis
ects
∠R
MO
an
d ∠
RH
O.
−−
−
RO
bis
ects
∠M
RH
an
d ∠
MO
H.
Exam
ple
60
16
24
90
32
3
10
4
Answers (Lesson 6-4 and Lesson 6-5)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Chapter 6 A15 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
32
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Rh
om
bi
an
d S
qu
are
s
Co
nd
itio
ns
for
Rh
om
bi
an
d S
qu
are
s T
he t
heore
ms
belo
w c
an
help
you
pro
ve t
hat
a p
ara
llelo
gra
m i
s a r
ect
an
gle
, rh
om
bu
s, o
r sq
uare
.
If t
he d
iagon
als
of
a p
ara
llelo
gra
m a
re p
erp
en
dic
ula
r, t
hen
th
e p
ara
llelo
gra
m i
s a r
hom
bu
s.
If o
ne d
iagon
al
of
a p
ara
llelo
gra
m b
isect
s a p
air
of
op
posi
te a
ngle
s, t
hen
th
e p
ara
llelo
gra
m
is a
rh
om
bu
s.
If o
ne p
air
of
con
secu
tive s
ides
of
a p
ara
llelo
gra
m a
re c
on
gru
en
t, t
he p
ara
llelo
gra
m i
s a
rhom
bu
s.
If a
qu
ad
rila
tera
l is
both
a r
ect
an
gle
an
d a
rh
om
bu
s, t
hen
it
is a
squ
are
.
D
ete
rm
ine
wh
eth
er p
ara
lle
log
ra
m ABCD
wit
h
ve
rti
ce
s A
(−3
, −
3),
B(1
, 1
), C
(5,
−3
), D
(1,
−7
) is
a rhombus,
a rectangle
, o
r a
square.
AC
= √
""
""
""
""
""
(-3 -
5)2
+ (
(-3 -
(-
3))
2 =
√ "
"
64 =
8
BD
= √
""
""
""
""
(1
-1)2
+ (
-7 -
1)2
= √
""
64 =
8
Th
e d
iagon
als
are
th
e s
am
e l
en
gth
; th
e f
igu
re i
s a r
ect
an
gle
.
Slo
pe o
f −
−
AC
=
-3 -
(-
3)
−
-3 -
5
=
0
−
-8 =
8
Th
e l
ine i
s h
ori
zon
tal.
Slo
pe o
f −−
− B
D =
1 -
(-
7)
−
1 -
1
=
8
−
0 =
un
def
ined
T
he l
ine i
s vert
ical.
Sin
ce a
hori
zon
tal
an
d v
ert
ical
lin
e a
re p
erp
en
dic
ula
r, t
he d
iagon
als
are
perp
en
dic
ula
r.
Para
llelo
gra
m A
BC
D i
s a s
qu
are
wh
ich
is
als
o a
rh
om
bu
s an
d a
rect
an
gle
.
Exerc
ises
Giv
en
ea
ch
se
t o
f v
erti
ce
s,
de
term
ine
wh
eth
er !
ABCD
is a
rhombus, rectangle
, o
r
square.
Lis
t a
ll t
ha
t a
pp
ly.
Ex
pla
in.
1. A
(0,
2),
B(2
, 4),
C(4
, 2),
D(2
, 0)
2. A
(-2,
1),
B(-
1,
3),
C(3
, 1),
D(2
, -
1)
3. A
(-2,
-1),
B(0
, 2),
C(2
, -
1),
D(0
, -
4)
4. A
(-3,
0),
B(-
1,
3),
C(5
, -
1),
D(3
, -
4)
5
. P
RO
OF W
rite
a t
wo-c
olu
mn
pro
of.
Giv
en
: P
ara
llelo
gra
m R
ST
U.
−−
RS
$ −
−
ST
Pro
ve
: R
ST
U i
s a r
hom
bu
s.
Sta
tem
en
ts
Re
as
on
s
1.
RS
TU
is
a p
ara
lle
log
ram
1
. G
ive
n
−
−
RS
$ −
−
ST
2. −
−
RS
$ −
−
UT , −
−
RU
$ −
−
ST
2.
De
fin
itio
n o
f a
pa
rall
elo
gra
m
3. −
−
UT $
−−
RS
$ −
−
ST $
−−
RU
3
. S
ub
sti
tuti
on
4.
RS
TU
is
a r
ho
mb
us
4
. D
efi
nit
ion
of
a r
ho
mb
us
6 -5 Exam
ple
y
x
R
ec
tan
gle
, rh
om
bu
s,
sq
ua
re;
the
fo
ur
sid
es
are
$ a
nd
co
ns
ec
uti
ve
s
ide
s a
re ⊥
.
R
ec
tan
gle
; c
on
se
cu
tiv
e s
ide
s a
re ⊥
.
R
ho
mb
us
; th
e f
ou
r s
ide
s a
re $
a
nd
co
ns
ec
uti
ve
sid
es
are
no
t ⊥
.
R
ec
tan
gle
; c
on
se
cu
tiv
e s
ide
s a
re ⊥
.
Lesson 6-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
33
G
len
co
e G
eo
me
try
Sk
ills
Pra
ctic
e
Rh
om
bi
an
d S
qu
are
s
ALG
EB
RA
Q
ua
dril
ate
ra
l DKLM
is a
rh
om
bu
s.
1
. If
DK
= 8
, fi
nd
KL
.
2
. If
m∠
DM
L=
82 f
ind
m∠
DK
M.
3
. If
m∠
KA
L=
2x
– 8,
fin
d x
.
4
. If
DA
= 4
x a
nd
AL
= 5
x–
3,
fin
d D
L.
5
. If
DA
= 4
x a
nd
AL
= 5
x–
3,
fin
d A
D.
6
. If
DM
= 5
y+
2 a
nd
DK
= 3
y+
6,
fin
d K
L.
7.P
RO
OF
Wri
te a
tw
o-c
olu
mn
pro
of.
Giv
en
:R
ST
U i
s a p
ara
llelo
gra
m.
−−−
RX
$−
−T
X $
−−
SX
$−−
−U
X
Pro
ve
:R
ST
U i
s a r
ect
an
gle
.P
ro
of
S
tate
me
nts
Re
aso
ns
1
.R
STU
is a
para
llelo
gra
m.
1.
Giv
en
−−
RX
$ −
−TX
$ −
−S
X $
−−
UX
2.
RX
= T
X =
SX
= U
X2.
Defi
nit
ion
of
co
ng
ruen
t seg
men
ts
3.
RX
+ X
T =
RT
, S
X +
XU
= S
U3.
Seg
Ad
d P
ostu
late
4.
RX
+X
T=
SU
4.
Su
bsti
tuti
on
Pro
pert
y
5.
RT =
SU
5.
Tra
nsit
ive P
rop
ert
y
6.
−−
RT $
−−
SU
6.
Defi
nit
ion
of
$ s
eg
men
t
7.
RS
TU
is a
recta
ng
le.
7.
If d
iag
on
als
$,
the f
igu
re i
s a
re
cta
ng
le.
CO
OR
DIN
AT
E G
EO
ME
TR
Y G
ive
n e
ach
se
t o
f v
erti
ce
s,
de
term
ine
wh
eth
er !
QRST
is a
rhombus,
a rectangle
, o
r a
square.
Lis
t a
ll t
ha
t a
pp
ly.
Ex
pla
in.
8. Q
(3,
5),
R(3
, 1),
S(-
1,
1),
T(-
1,
5)
9
. Q
(-5,
12),
R(5
, 12),
S(-
1,
4),
T(-
11,
4)
10
. Q
(-6,
-1),
R(4
, -
6),
S(2
, 5),
T(-
8,
10)
11
. Q
(2,
-4),
R(-
6,
-8),
S(-
10,
2),
T(-
2,
6)
ML
AD
K
6 -5
8
R
ho
mb
us,
recta
ng
le,
sq
uare
; all s
ides a
re c
on
gru
en
t an
d t
he
dia
go
nals
are
perp
en
dic
ula
r an
d c
on
gru
en
t.
R
ho
mb
us;
all s
ides a
re c
on
gru
en
t an
d t
he d
iag
on
als
are
perp
en
dic
ula
r,
bu
t n
ot
co
ng
ruen
t.
R
ho
mb
us;
all s
ides a
re c
on
gru
en
t an
d t
he d
iag
on
als
are
perp
en
dic
ula
r,
bu
t n
ot
co
ng
ruen
t.
N
on
e;
op
po
sit
e s
ides a
re c
on
gru
en
t, b
ut
the d
iag
on
als
are
neit
her
co
ng
ruen
t n
or
perp
en
dic
ula
r.
41
49
24
12
12
Answers (Lesson 6-5)
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Chapter 6 A16 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
34
G
len
co
e G
eo
me
try
Practi
ce
Rh
om
bi
an
d S
qu
are
s
PRYZ
is a
rh
om
bu
s.
If RK
= 5
, RY
= 1
3 a
nd
m∠YRZ
= 6
7,
fin
d e
ach
me
asu
re
.
1
. K
Y
2
. P
K
3
. m∠
YK
Z
4
. m∠
PZ
R
MNPQ
is a
rh
om
bu
s.
If PQ
= 3
√ #
2
an
d AP
= 3
, fi
nd
ea
ch
me
asu
re
.
5
. A
Q
6
. m∠
AP
Q
7
. m∠
MN
P
8
. P
M
CO
OR
DIN
AT
E G
EO
ME
TR
Y G
ive
n e
ach
se
t o
f v
erti
ce
s,
de
term
ine
wh
eth
er $
BEFG
is a
rhombus,
a rectangle
, o
r a
square.
Lis
t a
ll t
ha
t a
pp
ly.
Ex
pla
in.
9. B
(-9,
1),
E(2
, 3),
F(1
2, -
2),
G(1
, -
4)
10
. B
(1,
3),
E(7
, -
3),
F(1
, -
9),
G(-
5, -
3)
11
. B
(-4, -
5),
E(1
, -
5),
F(-
2, -
1),
G(-
7, -
1)
12
. T
ES
SE
LLA
TIO
NS
T
he f
igu
re i
s an
exam
ple
of
a t
ess
ell
ati
on
. U
se a
ru
ler
or
pro
tract
or
to m
easu
re t
he s
hap
es
an
d t
hen
nam
e t
he q
uad
rila
tera
ls
use
d t
o f
orm
th
e f
igu
re.
K
P
YR
Z
A
N MQP
6 -5
R
ho
mb
us;
all s
ides a
re c
on
gru
en
t an
d t
he d
iag
on
als
are
perp
en
dic
ula
r,
bu
t n
ot
co
ng
ruen
t.
R
ho
mb
us,
recta
ng
le,
sq
uare
; all s
ides a
re c
on
gru
en
t an
d t
he d
iag
on
als
are
perp
en
dic
ula
r an
d c
on
gru
en
t.
R
ho
mb
us;
all s
ides a
re c
on
gru
en
t an
d t
he d
iag
on
als
are
perp
en
dic
ula
r,
bu
t n
ot
co
ng
ruen
t.
T
he f
igu
re c
on
sis
ts o
f 6 c
on
gru
en
t rh
om
bi.
12
12
90
67
45 90
63
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 6-5
Ch
ap
ter
6
35
G
len
co
e G
eo
me
try
1. T
RA
Y R
AC
KS
A
tra
y r
ack
look
s li
ke a
p
ara
llelo
gra
m f
rom
th
e s
ide.
Th
e l
evels
fo
r th
e t
rays
are
even
ly s
pace
d.
Wh
at
two l
abele
d p
oin
ts f
orm
a r
hom
bu
s w
ith
base
AA' ?
2. S
LIC
ING
C
harl
es
cuts
a r
hom
bu
s alo
ng
both
dia
gon
als
. H
e e
nd
s u
p w
ith
fou
r co
ngru
en
t tr
ian
gle
s. C
lass
ify t
hese
tr
ian
gle
s as
acu
te,
obtu
se,
or
righ
t.
3. W
IND
OW
S T
he e
dges
of
a w
ind
ow
are
d
raw
n i
n t
he c
oord
inate
pla
ne.
Dete
rmin
e w
heth
er
the w
ind
ow
is
a
squ
are
or
a r
hom
bu
s.
4.S
QU
AR
ES
M
ack
en
zie
cu
t a s
qu
are
alo
ng i
ts d
iagon
als
to g
et
fou
r co
ngru
en
t ri
gh
t tr
ian
gle
s. S
he t
hen
join
ed
tw
o o
f th
em
alo
ng t
heir
lon
g s
ides.
Sh
ow
th
at
the r
esu
ltin
g s
hap
e i
s a s
qu
are
.
5. D
ES
IGN
T
ati
an
na m
ad
e t
he d
esi
gn
sh
ow
n.
Sh
e u
sed
32 c
on
gru
en
t rh
om
bi
to
create
th
e f
low
er-
lik
e d
esi
gn
at
each
co
rner.
a.
Wh
at
are
th
e a
ngle
s of
the c
orn
er
rhom
bi?
b.
Wh
at
kin
ds
of
qu
ad
rila
tera
ls a
re t
he
dott
ed
an
d c
heck
ere
d f
igu
res?
Wo
rd
Pro
ble
m P
racti
ce
Rh
om
bi
an
d S
qu
are
s
y
xO
AB
C
D
E
F
G
H
B,C,D,E,F,G,H,
A20 inches
AB
= 4
in
ch
es
,
6 -5
F a
nd
F′
rig
ht
tria
ng
les
rho
mb
us
Sam
ple
an
sw
er:
Th
e d
iag
on
als
of
a s
qu
are
cre
ate
fo
ur
co
ng
ruen
t
rig
ht
iso
scele
s t
rian
gle
s.
Wh
en
two
co
ng
ruen
t ri
gh
t is
oscele
s
tria
ng
les a
re j
oin
ed
alo
ng
th
eir
hyp
ote
nu
ses,
the r
esu
lt i
s a
qu
ad
rila
tera
l w
ith
4 e
qu
al
sid
es
an
d 4
rig
ht
an
gle
co
rners
(becau
se 4
5 +
45 =
90),
makin
g
it a
sq
uare
. 45,
135,
45,
135
Th
ey a
re a
ll s
qu
are
s.
Answers (Lesson 6-5)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Chapter 6 A17 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
36
G
len
co
e G
eo
me
try
En
rich
men
t
Cre
ati
ng
Pyth
ag
ore
an
Pu
zzle
s
By d
raw
ing t
wo s
qu
are
s an
d c
utt
ing t
hem
in
a c
ert
ain
way,
you
ca
n m
ak
e a
pu
zzle
th
at
dem
on
stra
tes
the P
yth
agore
an
Th
eore
m.
A s
am
ple
pu
zzle
is
show
n.
You
can
cre
ate
you
r ow
n p
uzzle
by
foll
ow
ing t
he i
nst
ruct
ion
s belo
w.
1
. C
are
full
y c
on
stru
ct a
squ
are
an
d l
abel
the l
en
gth
of
a s
ide a
s a
. T
hen
con
stru
ct a
sm
all
er
squ
are
to
the r
igh
t of
it a
nd
label
the l
en
gth
of
a s
ide a
s b,
as
show
n i
n t
he f
igu
re a
bove.
Th
e b
ase
s sh
ou
ld
be a
dja
cen
t an
d c
oll
inear.
2
. M
ark
a p
oin
t X
th
at
is b
un
its
from
th
e l
eft
ed
ge
of
the l
arg
er
squ
are
. T
hen
dra
w t
he s
egm
en
ts
from
th
e u
pp
er
left
corn
er
of
the l
arg
er
squ
are
to
poin
t X
, an
d f
rom
poin
t X
to t
he u
pp
er
righ
t co
rner
of
the s
mall
er
squ
are
.
3
. C
ut
ou
t an
d r
earr
an
ge y
ou
r fi
ve p
iece
s to
form
a
larg
er
squ
are
. D
raw
a d
iagra
m t
o s
how
you
r an
swer.
4
. V
eri
fy t
hat
the l
en
gth
of
each
sid
e i
s equ
al
to
√ "
""
a2 +
b2 .
a
a
bX
b
b
6 -5
See s
tud
en
ts’
wo
rk.
A s
am
ple
an
sw
er
is s
ho
wn
.
Lesson 6-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
37
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Tra
pezo
ids a
nd
Kit
es
Pro
pe
rtie
s o
f T
rap
ezo
ids
A t
ra
pe
zo
id i
s a q
uad
rila
tera
l w
ith
exact
ly o
ne p
air
of
para
llel
sid
es.
Th
e m
idse
gm
en
t or
me
dia
n
of
a t
rap
ezoid
is
the s
egm
en
t th
at
con
nect
s th
e m
idp
oin
ts o
f th
e l
egs
of
the t
rap
ezoid
. It
s m
easu
re i
s equ
al
to o
ne-h
alf
th
e s
um
of
the
len
gth
s of
the b
ase
s. I
f th
e l
egs
are
con
gru
en
t, t
he t
rap
ezoid
is
an
iso
sce
les t
ra
pe
zo
id.
In a
n i
sosc
ele
s tr
ap
ezoid
both
pair
s of
ba
se
an
gle
s a
re c
on
gru
en
t an
d t
he d
iagon
als
are
con
gru
en
t.
T
he
ve
rti
ce
s o
f ABCD
are
A(-
3,
-1
), B
(-1
, 3
),
C(2
, 3
), a
nd
D(4
, -
1).
Sh
ow
th
at ABCD
is a
tra
pe
zo
id a
nd
d
ete
rm
ine
wh
eth
er i
t is
an
iso
sce
les
tra
pe
zo
id.
slop
e o
f −
−
AB
=
3 -
(-
1)
−
-1 -
(-
3) =
4
−
2 =
2
slop
e o
f −−
− A
D =
-1 -
(-
1)
−
4 -
(-
3)
= 0
−
7 =
0
slop
e o
f −−
− B
C =
3 -
3
−
2 -
(-
1)
= 0
−
3 =
0
slop
e o
f −−
− C
D =
-1 -
3
−
4 -
2 =
-4
−
2 =
-2
Exact
ly t
wo s
ides
are
para
llel,
−−−
AD
an
d −−
− B
C ,
so A
BC
D i
s a t
rap
ezoid
. A
B =
CD
, so
AB
CD
is
an
iso
scele
s tr
ap
ezoid
.
Exerc
ises
Fin
d e
ach
me
asu
re
.
1
. m
∠D
55
2
. m
∠L
140
125°
5
5
40°
CO
OR
DIN
AT
E G
EO
ME
TR
Y F
or e
ach
qu
ad
ril
ate
ra
l w
ith
th
e g
ive
n v
erti
ce
s,
ve
rif
y
tha
t th
e q
ua
dril
ate
ra
l is
a t
ra
pe
zo
id a
nd
de
term
ine
wh
eth
er t
he
fig
ure
is a
n
iso
sce
les t
ra
pe
zo
id.
3
. A
(−1,
1),
B(3
, 2),
C(1
,−2),
D(−
2,
−1)
4. J
(1,
3),
K(3
, 1),
L(3
, −
2),
M(−
2,
3)
−
−
AD
‖ −
−
BC
, −
−
AB
∦ −
−
CD
, A
BC
D i
s a
−−
JK
‖ −
−
LM
, −
−
JM
∦ −
−
KL
, JK
LM
is a
n
trap
ezo
id b
ut
no
t an
iso
scele
s
iso
scele
s t
rap
ezo
id b
ecau
se
trap
ezo
id b
ecau
se A
B =
√ %
%
17 ,
J
M =
KL =
3.
CD
= √
%%
10 .
Fo
r t
ra
pe
zo
id HJKL
, M
an
d N
are
th
e m
idp
oin
ts o
f th
e l
eg
s.
5. I
f H
J =
32 a
nd
LK
= 6
0,
fin
d M
N.
46
6. If
HJ
= 1
8 a
nd
MN
= 2
8,
fin
d L
K.
38
T
RU
bas
e
bas
e
Sleg
x
y
O
B
DA
C
6-6
STU
R is a
n isoscele
s t
rapezoid
.
−−
SR
& −
−
TU
; ∠
R &
∠U
, ∠
S &
∠T
Exam
ple
AB
=
√ "
""
""
""
""
"
(-3 -
(-
1))
2 +
(-
1 -
3)2
=
√
""
"
4 +
16
= √
""
20 =
2 √
"
5
CD
=
√ "
""
""
""
""
(2
- 4
)2 +
(3 -
(-
1))
2
=
√ "
""
4 +
16 =
√ "
"
20 =
2 √
"
5
LK
NM
HJ
Answers (Lesson 6-5 and Lesson 6-6)
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Chapter 6 A18 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
38
G
len
co
e G
eo
me
try
Pro
pe
rtie
s o
f K
ite
s A
kit
e i
s a q
uad
rila
tera
l w
ith
exact
ly t
wo p
air
s of
con
secu
tive
con
gru
en
t si
des.
Un
lik
e a
para
llelo
gra
m,
the o
pp
osi
te s
ides
of
a k
ite a
re n
ot
con
gru
en
t or
para
llel.
Th
e d
iagon
als
of
a k
ite a
re p
erp
en
dic
ula
r.
For
kit
e R
MN
P,
−−
−
MP
⊥ −
−−
RN
In a
kit
e,
exact
ly o
ne p
air
of
op
posi
te a
ngle
s is
con
gru
en
t.
For
kit
e R
MN
P,
∠M
$ ∠
P
If
WX
YZ
is a
kit
e,
fin
d m
∠Z
.
Th
e m
easu
res
of
∠Y
an
d ∠
W a
re n
ot
con
gru
en
t, s
o ∠
X $
∠Z
.
m∠
X +
m∠
Y +
m∠
Z +
m∠
W =
360
m∠
X +
60
+ m
∠Z
+ 8
0 =
360
m∠
X +
m∠
Z =
220
m∠
X =
110, m
∠Z
= 1
10
If
AB
CD
is a
kit
e,
fin
d B
C.
Th
e d
iagon
als
of
a k
ite a
re p
erp
en
dic
ula
r. U
se t
he
Pyth
agore
an
Th
eore
m t
o f
ind
th
e m
issi
ng l
en
gth
.
BP
2 +
PC
2 =
BC
2
52 +
12
2 =
BC
2
169 =
BC
2
13 =
BC
Exerc
ises
If G
HJ
K i
s a
kit
e,
fin
d e
ach
me
asu
re
.
1. F
ind
m∠
JR
K.
90
2
. If
RJ
= 3
an
d R
K =
10,
fin
d J
K.
√ #
#
109
3
. If
m∠
GH
J =
90 a
nd
m∠
GK
J =
11
0,
fin
d m
∠H
GK
.
80
4
. If
HJ
= 7
, fi
nd
HG
.
7
5
. If
HG
= 7
an
d G
R =
5,
fin
d H
R.
√ #
#
24 =
2 √
#
6
6
. If
m∠
GH
J =
52 a
nd
m∠
GK
J =
95,
fin
d m
∠H
GK
.
106.5
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Tra
pezo
ids a
nd
Kit
es
6-6
Exam
ple
1
Exam
ple
2
80°
60°
5 5
4
P
12
Lesson 6-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
39
G
len
co
e G
eo
me
try
ALG
EB
RA
F
ind
ea
ch
me
asu
re
.
1
. m
∠S
117
2
. m
∠M
38
63°
14
14
142°
21
21
3
. m
∠D
127
4
. R
H
√ #
#
544 =
4 √
##
34
36°
70°
20
12
12
ALG
EB
RA
Fo
r t
ra
pe
zo
id H
JK
L,
T a
nd
S a
re
mid
po
ints
of
the
le
gs.
5
. If
HJ
= 1
4 a
nd
LK
= 4
2,
fin
d T
S.
28
6
. If
LK
= 1
9 a
nd
TS
= 1
5,
fin
d H
J.
11
7
. If
HJ
= 7
an
d T
S =
10,
fin
d L
K.
13
8
. If
KL
= 1
7 a
nd
JH
= 9
, fi
nd
ST
.
13
CO
OR
DIN
AT
E G
EO
ME
TR
Y E
FG
H i
s a
qu
ad
ril
ate
ra
l w
ith
ve
rti
ce
s E
(1,
3),
F(5
, 0
),
G(8
, −
5),
H(−
4,
4).
9
. V
eri
fy t
hat
EF
GH
is
a t
rap
ezoid
.
−−
EF
‖ −
−
GH
, bu
t −
−
HE
∦ −
−
FG
10
. D
ete
rmin
e w
heth
er
EF
GH
is
an
iso
scele
s tr
ap
ezoid
. E
xp
lain
.
n
ot
iso
scele
s;
EH
= √
##
26 a
nd
FG
= √
##
34
6-6
Sk
ills
Pra
ctic
e
Tra
pezo
ids a
nd
Kit
es
Answers (Lesson 6-6)
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Chapter 6 A19 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
40
G
len
co
e G
eo
me
try
Practi
ce
Tra
pezo
ids a
nd
Kit
es
Fin
d e
ach
me
asu
re
.
1
. m∠T
60
2
. m∠Y
112
60°
68°
7 7
3
. m∠Q
101
4
. BC
√
""
65
110°
48°
11
4
7 7
ALG
EB
RA
F
or t
ra
pe
zo
id FEDC
, V
an
d Y
are
mid
po
ints
of
the
le
gs.
5
. If
FE
= 1
8 a
nd
VY
= 2
8,
fin
d CD
. 38
6
. If
m∠F
= 1
40 a
nd
m∠E
= 1
25,
fin
d m∠D
. 55
CO
OR
DIN
AT
E G
EO
ME
TR
Y RSTU
is a
qu
ad
ril
ate
ra
l w
ith
ve
rti
ce
s R
(−3
, −
3),
S(5
, 1
),
T(1
0,
−2
), U
(−4
, −
9).
7
. V
eri
fy t
hat RSTU
is
a t
rap
ezoid
.
−−
RS
‖ −
−
TU
8
. D
ete
rmin
e w
heth
er RSTU
is
an
iso
scele
s tr
ap
ezoid
. E
xp
lain
.
n
ot
iso
scele
s;
RU
= √
""
37 a
nd
ST
= √
""
34
9
. C
ON
ST
RU
CT
ION
A
set
of
stair
s le
ad
ing t
o t
he e
ntr
an
ce o
f a b
uil
din
g i
s d
esi
gn
ed
in
th
e
shap
e o
f an
iso
scele
s tr
ap
ezoid
wit
h t
he l
on
ger
base
at
the b
ott
om
of
the s
tair
s an
d t
he
short
er
base
at
the t
op
. If
th
e b
ott
om
of
the s
tair
s is
21 f
eet
wid
e a
nd
th
e t
op
is
14 f
eet
wid
e,
fin
d t
he w
idth
of
the s
tair
s h
alf
way t
o t
he t
op
.
10
. D
ES
K T
OP
S A
carp
en
ter
need
s to
rep
lace
severa
l tr
ap
ezoid
-sh
ap
ed
desk
top
s in
a
class
room
. T
he c
arp
en
ter
kn
ow
s th
e l
en
gth
s of
both
base
s of
the d
esk
top
. W
hat
oth
er
measu
rem
en
ts,
if a
ny,
does
the c
arp
en
ter
need
?
6-6
FE
DC
VY
17.5
ft
Sam
ple
an
sw
er:
th
e m
easu
res o
f th
e b
ase a
ng
les
Lesson 6-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
41
G
len
co
e G
eo
me
try
Wo
rd
Pro
ble
m P
racti
ce
Tra
pezo
ids a
nd
Kit
es
1
. P
ER
SP
EC
TIV
E A
rtis
ts u
se d
iffe
ren
t
tech
niq
ues
to m
ak
e t
hin
gs
ap
pear
to b
e
3-d
imen
sion
al
wh
en
dra
win
g i
n t
wo
dim
en
sion
s. K
evin
dre
w t
he w
all
s of
a
room
. In
real
life
, all
of
the w
all
s are
rect
an
gle
s. I
n w
hat
shap
e d
id h
e d
raw
the s
ide w
all
s to
mak
e t
hem
ap
pear
3-d
imen
sion
al?
2.P
LA
ZA
In
ord
er
to g
ive t
he f
eeli
ng o
f
spaci
ou
sness
, an
arc
hit
ect
deci
des
to
mak
e a
pla
za i
n t
he s
hap
e o
f a k
ite.
Th
ree o
f th
e f
ou
r co
rners
of
the p
laza
are
sh
ow
n o
n t
he c
oord
inate
pla
ne.
If
the f
ou
rth
corn
er
is i
n t
he f
irst
qu
ad
ran
t, w
hat
are
its
coord
inate
s?
y
x
y
x
(6,
1)
3
. A
IRP
OR
TS
A
sim
pli
fied
dra
win
g o
f
the r
eef
run
way c
om
ple
x a
t H
on
olu
lu
Inte
rnati
on
al
Air
port
is
show
n b
elo
w.
How
man
y t
rap
ezoid
s are
th
ere
in
th
is
image?
4
. LIG
HT
ING
A
lig
ht
ou
tsid
e a
room
sh
ines
thro
ugh
th
e d
oor
an
d i
llu
min
ate
s a
trap
ezoid
al
regio
n ABCD
on
th
e f
loor.
AB
DC
Un
der
wh
at
circ
um
stan
ces
wou
ld
trap
ezoid
ABCD
be i
sosc
ele
s?
5. R
ISE
RS
A
ris
er
is d
esi
gn
ed
to e
levate
a s
peak
er.
Th
e r
iser
con
sist
s of
4 t
rap
ezoid
al
sect
ion
s th
at
can
be
stack
ed
on
e o
n t
op
of
the o
ther
to
pro
du
ce t
rap
ezoid
s of
vary
ing h
eig
hts
.
20 feet
10 feet
All
of
the s
tages
have t
he s
am
e h
eig
ht.
If a
ll f
ou
r st
ages
are
use
d,
the w
idth
of
the t
op
of
the r
iser
is 1
0 f
eet.
a.
If o
nly
th
e b
ott
om
tw
o s
tages
are
use
d,
wh
at
is t
he w
idth
of
the t
op
of
the r
esu
ltin
g r
iser?
b.
Wh
at
wou
ld b
e t
he w
idth
of
the r
iser
if t
he b
ott
om
th
ree s
tages
are
use
d?
6-6 tr
ap
ezo
ids
W
hen
th
e l
igh
t so
urc
e i
s a
n e
qu
al
dis
tan
ce f
rom
C a
nd
D,
sh
inin
g
str
aig
ht
thro
ug
h t
he d
oo
r.
15 f
t
12.5
ft5
Answers (Lesson 6-6)
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Chapter 6 A20 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
6
42
G
len
co
e G
eo
me
try
En
ric
hm
en
t
Qu
ad
rila
tera
ls i
n C
on
str
ucti
on
Qu
ad
rila
tera
ls a
re o
ften
use
d i
n c
on
stru
ctio
n w
ork
.
1. T
he d
iagra
m a
t th
e r
igh
t re
pre
sen
ts a
roof
fr
am
e a
nd
sh
ow
s m
an
y q
uad
rila
tera
ls.
Fin
d
the f
oll
ow
ing s
hap
es
in t
he d
iagra
m a
nd
sh
ad
e i
n t
heir
ed
ges.
a.
isosc
ele
s tr
ian
gle
b.
scale
ne t
rian
gle
c.
rect
an
gle
d.
rhom
bu
s
e.
trap
ezoid
(n
ot
isosc
ele
s)
f.
isosc
ele
s tr
ap
ezoid
2. T
he f
igu
re a
t th
e r
igh
t re
pre
sen
ts a
win
dow
. T
he
wood
en
part
betw
een
th
e p
an
es
of
gla
ss i
s 3 i
nch
es
wid
e.
Th
e f
ram
e a
rou
nd
th
e o
ute
r ed
ge i
s 9 i
nch
es
wid
e.
Th
e o
uts
ide m
easu
rem
en
ts o
f th
e f
ram
e a
re
60 i
nch
es
by 8
1 i
nch
es.
Th
e h
eig
ht
of
the t
op
an
d
bott
om
pan
es
is t
he s
am
e.
Th
e t
op
th
ree p
an
es
are
th
e s
am
e s
ize.
a.
How
wid
e i
s th
e b
ott
om
pan
e o
f gla
ss?
b.
How
wid
e i
s each
top
pan
e o
f gla
ss?
c.
How
hig
h i
s each
pan
e o
f gla
ss?
3. E
ach
ed
ge o
f th
is b
ox h
as
been
rein
forc
ed
w
ith
a p
iece
of
tap
e.
Th
e b
ox i
s 10 i
nch
es
hig
h,
20 i
nch
es
wid
e,
an
d 1
2 i
nch
es
deep
. W
hat
is t
he l
en
gth
of
the t
ap
e t
hat
has
been
use
d?
jack
raf
ters
hip
raf
ter
com
mon r
afte
rs
Ro
of
Fra
me
ridge
boar
dva
lley
raft
er
pla
te
60 in
.
81 in
.
9 in
.
3 in
.
3 in
.
9 in
.
9 in
.
10 in
.
12 in
.
20 in
.
6-6
See s
tud
en
ts’
wo
rk.
42 i
n.
12 i
n.
30 i
n.
168 i
n.
Answers (Lesson 6-6)
Copyright
© G
lencoe/M
cG
raw
-Hill
, a d
ivis
ion o
f T
he M
cG
raw
-Hill
Com
panie
s,
Inc.
Answers
Chapter 6 A21 Glencoe Geometry
Chapter 6 Assessment Answer Key
Quiz 2 (Lesson 6-3)
Page 45
Quiz 1 (Lessons 6-1 and 6-2) Quiz 3 (Lessons 6-4 and 6-5) Mid-Chapter Test
Page 45 Page 46 Page 47
Quiz 4 (Lesson 6-6)
Page 46
1.
2.
3.
4.
5.
12,240
45
20
(-1, 10)
A
1.
2.
3.
4.
5.
No; none of the
tests for s are ful! lled.
true
false
true
28 cm
1.
2.
3.
4.
5.
B
65
115
true
rectangle, rhombus, square
1.
2.
3.
4.
5.
118
√ ## 50 = 5 √ # 2
5
15.5
Use the distance formula to show CF = DE.
1.
2.
3.
4.
5.
B
J
A
G
C
6.
7.
8.
9.
10.
50
42
yes; opp. sides are $ to each other
30, 150
No; slope −−
XY = - 3 −
5
and slope of −−
WZ = - 1 −
3 ,
so opposite sides are
not parallel.
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Chapter 6 A22 Glencoe Geometry
Chapter 6 Assessment Answer KeyVocabulary Test Form 1
Page 48 Page 49 Page 50
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
isosceles trapezoid
parallelogram
trapezoid
square
rhombus
false, rectangle
false; rhombus
diagonals
median
Sample answer:
angles formed by the
base and one of the
legs of the trapezoid
the nonparallel sides of a trapezoid
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
B
H
B
H
D
F
B
F
D
F
C
12.
13.
14.
15.
16.
17.
18.
19.
20.
H
B
J
A
G
A
J
A
G
B:
x = 7, m∠WYZ = 41
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Chapter 6 A23 Glencoe Geometry
Chapter 6 Assessment Answer KeyForm 2A Form 2B
Page 51 Page 52 Page 53 Page 54
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
B
G
D
F
C
F
A
H
B
F
D
12.
13.
14.
15.
16.
17.
18.
19.
20.
J
C
H
A
G
D
H
D
G
B: 7 or -4
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
B
J
D
F
B
G
A
H
C
J
B
12.
13.
14.
15.
16.
17.
18.
19.
20.
G
D
G
D
H
C
F
A
H
B: 22
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Chapter 6 A24 Glencoe Geometry
Chapter 6 Assessment Answer KeyForm 2C
Page 55 Page 56
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
1080
19
40
18
8
122
(6, 4)
Yes; −−
AB and −−
CD are
|| and ".
No; the slopes are
9 −
4 ,
1 −
7 , 1, and
2 −
3 .
Thus, ABCD doesnot have || sides.
- 2 − 3
22
Yes; if the diagonals
of a # are ", then
the # is a rectangle.
67.5
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
B:
(4, 0)
31
Yes; ABCD has only one pair of opposite sides ||, −−
BC and −−
AD .
6
26
true
true
true
false
true
true
false
x = 9, y = 2
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Chapter 6 A25 Glencoe Geometry
Chapter 6 Assessment Answer KeyForm 2D
Page 57 Page 58
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
360
38
90
50
3.6
117
( 1 − 2 , 3)
Yes; Both pairs of
opp. sides are ".
ABCD has two pairs of
|| sides, −−
AB || −−
CD
and −−
BD || −−
DA ; it is a #.
-4
8
One rt. ∠ means that the other % will be rt. %. If all 4 % are
rt. %, the & is a rectangle.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
B:
72
(-3, 1)
16
Yes; ABCD has only
one pair of opp.
sides ||, −−
AB and −−
BD .
8
17
false
true
false
false
true
false
true
90
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Chapter 6 A26 Glencoe Geometry
Chapter 6 Assessment Answer KeyForm 3
Page 59 Page 60
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
7
30; 30, 47, 120,
179, 174, and 170
180
− x
65
8 or 32
Yes; the diagonals bisect each other.
slope of −−
CD = 2 −
3 ;
slope of −−
DA = -2.
28
72
8 √ # 2
4
9
13
14.
15.
16.
17.
18.
19.
20.
B:
No; two pairs of congruent consecutive
sides do not exist.
CD = √ ## 72 , DA = √ ## 65 ,
AB = √ ## 20 , BC = √ ## 17
Yes; −−
AB ⊥ −−
BC ,
−− BC ⊥
−− CD ,
−− CD ⊥
−− AD ,
so opp % are &, and
all % are rt. %.
ABCD has 2 pairs of
opp. sides &, −−
AB
& −−
CD and −−
BC & −−
DA ,
so ABCD is a '.
105
Opp. sides of a ' are &.
If both pairs of opp. sides of a
quad. are &, then the quad.
is a '.
27
54
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Chapter 6 A27 Glencoe Geometry
Chapter 6 Assessment Answer Key
Extended-Response Test, Page 61
Scoring Rubric
Score General Description Speci! c Criteria
4 Superior
A correct solution that
is supported by well-
developed, accurate
explanations
• Shows thorough understanding of the concepts of angles of
polygons, properties of parallelograms, rectangles, rhombi,
squares, and trapezoids.
• Uses appropriate strategies to solve problems.
• Computations are correct.
• Written explanations are exemplary.
• Graphs and " gures are accurate and appropriate.
• Goes beyond requirements of some or all problems.
3 Satisfactory
A generally correct solution,
but may contain minor # aws
in reasoning or computation
• Shows an understanding of the concepts of angles of polygons,
properties of parallelograms, rectangles, rhombi, squares, and
trapezoids.
• Uses appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are effective.
• Graphs and " gures are mostly accurate and appropriate.
• Satis" es all requirements of problems.
2 Nearly Satisfactory
A partially correct
interpretation and/or solution
to the problem
• Shows an understanding of most of the concepts of angles of
polygons, properties of parallelograms, rectangles, rhombi,
squares, and trapezoids.
• May not use appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are satisfactory.
• Graphs and " gures are mostly accurate.
• Satis" es the requirements of most of the problems.
1 Nearly Unsatisfactory
A correct solution with no
supporting evidence or
explanation
• Final computation is correct.
• No written explanations or work shown to substantiate the " nal
computation.
• Graphs and " gures may be accurate but lack detail or explanation.
• Satis" es minimal requirements of some of the problems.
0 Unsatisfactory
An incorrect solution
indicating no mathematical
understanding of the concept
or task, or no solution is
given
• Shows little or no understanding of most of the concepts of angles
of polygons, properties of parallelograms, rectangles, rhombi,
squares, and trapezoids.
• Does not use appropriate strategies to solve problems.
• Computations are incorrect.
• Written explanations are unsatisfactory.
• Graphs and " gures are inaccurate or inappropriate.
• Does not satisfy requirements of problems.
• No answer may be given.
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Chapter 6 A28 Glencoe Geometry
Chapter 6 Assessment Answer Key Extended-Response Test, Page 61
Sample Answers
1. a. Any type of convex polygon can be drawn as long as one is regular and one is not regular and both have the same number of sides.
90°
70°
60°
110°120°
90°
90°
90°
b. Check to be sure that the exterior angles have been properly drawn and accurately measured.
c. 4(90) = 360; 120 + 70 + 60 + 110 = 360; The sum of the exterior angles of each figure should be 360°. The sum of the exterior angles of both the regular convex polygon and the irregular convex polygon is 360°.
2. The student should draw a rectangle and join the midpoints of consecutive sides. The figure formed inside is a rhombus. Since all four small triangles can be proved to be congruent by SAS, the four sides of the interior quadrilateral are congruent by CPCTC, making it a rhombus.
3. The student should draw an isosceles trapezoid with one pair of opposite sides parallel and the other pair of opposite sides congruent, as in the figure below.
4. a. Possible properties:A square has four congruent sides and a rectangle may not.A square has perpendicular diagonals and a rectangle may not.The diagonals of a square bisect the angles and those in a rectangle may not.
b. Possible properties:A square has four right angles and a rhombus may not.The diagonals of a square are congruent and those of a rhombus may not be.
c. Possible properties:A rectangle has four right angles and a parallelogram may not.The diagonals of a rectangle are congruent and those of a parallelogram may not be.
In addition to the scoring rubric found on page A30, the following sample answers
may be used as guidance in evaluating open-ended assessment items.
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Chapter 6 A29 Glencoe Geometry
Chapter 6 Assessment Answer KeyStandardized Test Practice
Page 62 Page 63
8. F G H J
9. A B C D
10. F G H J
11. A B C D
12. F G H J
13.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
7
14.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9 0 0
1. A B C D
2. F G H J
3. A B C D
4. F G H J
5. A B C D
6. F G H J
7. A B C D
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Chapter 6 A30 Glencoe Geometry
Chapter 6 Assessment Answer KeyStandardized Test Practice (continued)
Page 64
15.
16.
17.
18.
19.
20.
21. a.
b.
c.
hexagon;concave;irregular
88.9 cm
−−
DL
x = 11, y = 2
Assume thatneither bell costmore than $45.
40 cm3
−−
ML
−−
JL
J
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Chapter 6 A31 Glencoe Geometry
Chapter 6 Assessment Answer KeyUnit 2 Test
Page 65 Page 66
11.
12.
13.
14.
15.
16.
17.
18.
19.
yes
7 − 2 < x < 31
− 2
9
m∠JHK = 52;
m∠HMK = 108,
and x = 8
No; opp. side are
not ||.
5
5
11
In a parallelogram,
opposite sides are
congruent. Using
the distance
formula, PQ = √ ## 20 ,
PS = √ ## 20 , QR = √ # 8 ,RS = √ # 8 .
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
right; obtuse;
acute
m∠1 = 25; m∠2 = 25; m∠3 = 130
−−
AF % −−
ST , −−
FP % −−
TX , −−
PA % −−
XS
yes
ASA Postulate
∠A % ∠C
111
a = 9;
m∠ZWT = 32
∠YWZ
Neither appliance cost
more than $603.
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