Post on 27-Apr-2018
Chapter 3Resource Masters
Geometry
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3
ANSWERS FOR WORKBOOKS The answers for Chapter 3 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. 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’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-860180-0 GeometryChapter 3 Resource Masters
1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03
© Glencoe/McGraw-Hill iii Glencoe Geometry
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Proof Builder . . . . . . . . . . . . . . . . . . . . . . ix
Lesson 3-1Study Guide and Intervention . . . . . . . . 125–126Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 127Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Reading to Learn Mathematics . . . . . . . . . . 129Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 130
Lesson 3-2Study Guide and Intervention . . . . . . . . 131–132Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 133Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Reading to Learn Mathematics . . . . . . . . . . 135Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 136
Lesson 3-3Study Guide and Intervention . . . . . . . . 137–138Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 139Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Reading to Learn Mathematics . . . . . . . . . . 141Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 142
Lesson 3-4Study Guide and Intervention . . . . . . . . 143–144Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 145Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Reading to Learn Mathematics . . . . . . . . . . 147Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 148
Lesson 3-5Study Guide and Intervention . . . . . . . . 149–150Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 151Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Reading to Learn Mathematics . . . . . . . . . . 153Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 154
Lesson 3-6Study Guide and Intervention . . . . . . . . 155–156Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 157Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 158Reading to Learn Mathematics . . . . . . . . . . 159Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 160
Chapter 3 AssessmentChapter 3 Test, Form 1 . . . . . . . . . . . . 161–162Chapter 3 Test, Form 2A . . . . . . . . . . . 163–164Chapter 3 Test, Form 2B . . . . . . . . . . . 165–166Chapter 3 Test, Form 2C . . . . . . . . . . . 167–168Chapter 3 Test, Form 2D . . . . . . . . . . . 169–170Chapter 3 Test, Form 3 . . . . . . . . . . . . 171–172Chapter 3 Open-Ended Assessment . . . . . . 173Chapter 3 Vocabulary Test/Review . . . . . . . 174Chapter 3 Quizzes 1 & 2 . . . . . . . . . . . . . . . 175Chapter 3 Quizzes 3 & 4 . . . . . . . . . . . . . . . 176Chapter 3 Mid-Chapter Test . . . . . . . . . . . . 177Chapter 3 Cumulative Review . . . . . . . . . . . 178Chapter 3 Standardized Test Practice . 179–180Unit 1 Test/Review (Ch. 1–3) . . . . . . . . 181–182
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A30
© Glencoe/McGraw-Hill iv Glencoe Geometry
Teacher’s Guide to Using theChapter 3 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 3 Resource Masters includes the core materials neededfor Chapter 3. 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 in theGeometry TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 3-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.
Vocabulary Builder Pages ix–xinclude another student study tool thatpresents up to fourteen of the key theoremsand postulates from the chapter. Studentsare to write each theorem or postulate intheir own words, including illustrations ifthey choose to do so. You may suggest thatstudents highlight or star the theorems orpostulates with which they are not familiar.
WHEN TO USE Give these pages tostudents before beginning Lesson 3-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toupdate it as they complete each lesson.
Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
© Glencoe/McGraw-Hill v Glencoe Geometry
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
Assessment OptionsThe assessment masters in the Chapter 3Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 172–173. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
33
© Glencoe/McGraw-Hill vii Glencoe Geometry
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 3.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
alternate exterior angles
alternate interior angles
consecutive interior angles
corresponding angles
equidistant
ee·kwuh·DIS·tuhnt
non-Euclidean geometry
yoo·KLID·ee·yuhn
parallel lines
parallel planes
(continued on the next page)
© Glencoe/McGraw-Hill viii Glencoe Geometry
Vocabulary Term Found on Page Definition/Description/Example
plane Euclidean geometry
point-slope form
rate of change
skew lines
slope
slope-intercept form
spherical geometry
SFIR·uh·kuhl
transversal
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
33
Learning to Read MathematicsProof Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
33
© Glencoe/McGraw-Hill ix Glencoe Geometry
Proo
f Bu
ilderThis is a list of key theorems and postulates you will learn in Chapter 3. As you
study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 3.1Alternate Interior Angles Theorem
Theorem 3.2Consecutive Interior Angles Theorem
Theorem 3.3Alternate Exterior Angles Theorem
Theorem 3.4Perpendicular Transversal Theorem
Theorem 3.5
Theorem 3.6
Theorem 3.7
(continued on the next page)
© Glencoe/McGraw-Hill x Glencoe Geometry
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 3.8
Theorem 3.9
Postulate 3.1Corresponding Angles Postulate
Postulate 3.2
Postulate 3.3
Postulate 3.4
Postulate 3.5Parallel Postulate
Learning to Read MathematicsProof Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
33
Study Guide and InterventionParallel Lines and Transversals
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
© Glencoe/McGraw-Hill 125 Glencoe Geometry
Less
on
3-1
Relationships Between Lines and Planes When two lines lie in the same plane and do not intersect, they are parallel.Lines that do not intersect and are not coplanar are skew lines.In the figure, � is parallel to m, or � || m. You can also write PQ��� || RS���. Similarly, if two planes do not intersect, they are parallel planes.
a. Name all planes that are parallel to plane ABD.plane EFH
b. Name all segments that are parallel to C�G�.B�F�, D�H�, and A�E�
c. Name all segments that are skew to E�H�.B�F�, C�G�, B�D�, C�D�, and A�B�
For Exercises 1–3, refer to the figure at the right.
1. Name all planes that intersect plane OPT.
2. Name all segments that are parallel to N�U�.
3. Name all segments that intersect M�P�.
For Exercises 4–7, refer to the figure at the right.
4. Name all segments parallel to Q�X�.
5. Name all planes that intersect plane MHE.
6. Name all segments parallel to Q�R�.
7. Name all segments skew to A�G�.
T
SG
A
X
EN
MH
O
Q
R
M
N O
P
R
U T
S
A
B C
D
E
F G
H
P Q
R S
n
m
�
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 126 Glencoe Geometry
Angle Relationships A line that intersects two or more other lines in a plane is calleda transversal. In the figure below, t is a transversal. Two lines and a transversal form eightangles. Some pairs of the angles have special names. The following chart lists the pairs ofangles and their names.
Angle Pairs Name
�3, �4, �5, and �6 interior angles
�3 and �5; �4 and �6 alternate interior angles
�3 and �6; �4 and �5 consecutive interior angles
�1, �2, �7, and �8 exterior angles
�1 and �7; �2 and �8 alternate exterior angles
�1 and �5; �2 and �6; �3 and �7; �4 and �8
corresponding angles
Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutiveinterior angles.
a. �10 and �16 b. �4 and �12alternate exterior angles corresponding angles
c. �12 and �13 d. �3 and �9consecutive interior angles alternate interior angles
Use the figure in the Example for Exercises 1–12.
Name the transversal that forms each pair of angles.
1. �9 and �13 2. �5 and �14 3. �4 and �6
Identify each pair of angles as alternate interior, alternate exterior,corresponding, or consecutive interior angles.
4. �1 and �5 5. �6 and �14 6. �2 and �8
7. �3 and �11 8. �12 and �3 9. �4 and �6
10. �6 and �16 11. �11 and �14 12. �10 and �16
�
p q
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141365151678
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n
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34
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Study Guide and Intervention (continued)
Parallel Lines and Transversals
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
ExampleExample
ExercisesExercises
Skills PracticeParallel Lines and Transversals
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
© Glencoe/McGraw-Hill 127 Glencoe Geometry
Less
on
3-1
For Exercises 1–4, refer to the figure at the right.
1. Name all planes that are parallel to plane DEH.
2. Name all segments that are parallel to A�B�.
3. Name all segments that intersect G�H�.
4. Name all segments that are skew to C�D�.
Identify the sets of lines to which each given line is a transversal.
5. r
6. s
7. w
Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles.
8. �2 and �8 9. �3 and �6
10. �1 and �9 11. �3 and �9
12. �6 and �12 13. �7 and �11
Name the transversal that forms each pair of angles. Then identify the special name for the angle pair.
14. �4 and �10 15. �2 and �12
16. �7 and �3 17. �13 and �10
18. �8 and �14 19. �6 and �14
h
g
j
k
12
910
1112
13 1514
1634
65 7
8
a
b
cd
1 2
9 101112
3465
78
rs
t
w
z
A
E F
B
D
H G
C
© Glencoe/McGraw-Hill 128 Glencoe Geometry
For Exercises 1–4, refer to the figure at the right.
1. Name all planes that intersect plane STX.
2. Name all segments that intersect Q�U�.
3. Name all segments that are parallel to X�Y�.
4. Name all segments that are skew to V�W�.
Identify the sets of lines to which each given line is a transversal.
5. e
6. h
Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles.
7. �9 and �13 8. �6 and �16
9. �3 and �10 10. �8 and �14
Name the transversal that forms each pair of angles. Then identify the special name for the angle pair.
11. �2 and �12 12. �6 and �18
13. �13 and �19 14. �11 and �7
FURNITURE For Exercises 15–16, refer to the drawing of the end table.
15. Find an example of parallel planes.
16. Find an example of parallel lines.
cd
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a1
9 101112
13 141516
1719
1820
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Y
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Practice Parallel Lines and Transversals
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
Reading to Learn MathematicsParallel Lines and Transversals
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
© Glencoe/McGraw-Hill 129 Glencoe Geometry
Less
on
3-1
Pre-Activity How are parallel lines and planes used in architecture?
Read the introduction to Lesson 3-1 at the top of page 126 in your textbook.
• Give an example of parallel lines that can be found in your classroom.
• Give an example of parallel planes that can be found in your classroom.
Reading the Lesson
1. Write a geometrical term that matches each definition.
a. two planes that do not intersect
b. lines that are not coplanar and do not intersect
c. two coplanar lines that do not intersect
d. a line that intersects two or more lines in a plane at different points
e. a pair of angles determined by two lines and a transversal consisting of an interiorangle and an exterior angle that have different vertices and that lie on the same sideof the transversal
2. Refer to the figure at the right. Give the special name for each angle pair.
a. �3 and �5
b. �6 and �12
c. �4 and �8
d. �2 and �3
e. �8 and �12
f. �5 and �9
g. �4 and �10
h. �6 and �7
Helping You Remember
3. A good way to remember new mathematical terms is to relate them to words that youuse in everyday life. Many words start with the prefix trans-, which is a Latin rootmeaning across. List four English words that start with trans-. How can the meaning ofthis prefix help you remember the meaning of transversal?
�m
np
12
910
1112
34
65
78
© Glencoe/McGraw-Hill 130 Glencoe Geometry
Perspective DrawingsTo draw three-dimensional objects, artists makeperspective drawingssuch as the ones shown.To indicate depth in aperspective drawing, someparallel lines are drawn as converging lines. Thedotted lines in the figuresbelow each extend to avanishing point, or spotwhere parallel lines appear to meet.
Draw lines to locate the vanishing point in each drawing of a box.
1. 2. 3.
4. The fronts of two cubes are shown below. Using point P as the vanishing point for both cubes, complete the perspective drawings of the cubes.
5. Find an example of a perspective drawing in a newspaper or magazine.Trace the drawing and locate a vanishing point.
P
Vanishing points
Railroad tracks Cube Cabinet
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
Study Guide and InterventionAngles and Parallel Lines
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
© Glencoe/McGraw-Hill 131 Glencoe Geometry
Less
on
3-2
Parallel Lines and Angle Pairs When two parallel lines are cut by a transversal,the following pairs of angles are congruent.
• corresponding angles• alternate interior angles• alternate exterior angles
Also, consecutive interior angles are supplementary.
In the figure, m�2 � 75. Find the measures of the remaining angles.m�1 � 105 �1 and �2 form a linear pair.m�3 � 105 �3 and �2 form a linear pair.m�4 � 75 �4 and �2 are vertical angles.m�5 � 105 �5 and �3 are alternate interior angles.m�6 � 75 �6 and �2 are corresponding angles.m�7 � 105 �7 and �3 are corresponding angles.m�8 � 75 �8 and �6 are vertical angles.
In the figure, m�3 � 102. Find the measure of each angle.
1. �5 2. �6
3. �11 4. �7
5. �15 6. �14
In the figure, m�9 � 80 and m�5 � 68. Find the measure of each angle.
7. �12 8. �1
9. �4 10. �3
11. �7 12. �16
w v
p
q
1 234
6578
9 101112
1413
1516
p q
m
n
1 234
6578
9 101112
14131516
p
m
n
1 234
6578
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 132 Glencoe Geometry
Algebra and Angle Measures Algebra can be used to find unknown values in angles formed by a transversal and parallel lines.
If m�1 � 3x � 15, m�2 � 4x � 5, m�3 � 5y,and m�4 � 6z � 3, find x and y.
p || q, so m�1 � m�2 because they are corresponding angles.
3x � 15 � 4x � 53x � 15 � 3x � 4x � 5 � 3x
15 � x � 515 � 5 � x � 5 � 5
20 � x
p q
r
s
1 2
34
Study Guide and Intervention (continued)
Angles and Parallel Lines
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
ExampleExample
r || s, so m�2 � m�3 because they are corresponding angles.
m�2 � m�375 � 5y
�755� � �
55y�
15 � y
ExercisesExercises
Find x and y in each figure.
1. 2.
3. 4.
Find x, y, and z in each figure.
5. 6.
2x�
2y�
90� x�
z�
2y�106�
x�
(4z � 6)�
(5x � 20)�
3x�2y �
4y �
(11x � 4)�
(13y � 5)�
(5y � 5)�
5x�
(15x � 30)�
(3y � 18)� 10x�
90�
(5x � 5)�(6y � 4)�
(4x � 10)�
Skills PracticeAngles and Parallel Lines
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
© Glencoe/McGraw-Hill 133 Glencoe Geometry
Less
on
3-2
In the figure, m�2 � 70. Find the measure of each angle.
1. �3 2. �5
3. �8 4. �1
5. �4 6. �6
In the figure, m�7 � 100. Find the measure of each angle.
7. �9 8. �6
9. �8 10. �2
11. �5 12. �11
In the figure, m�3 � 75 and m�10 � 115. Find the measure of each angle.
13. �2 14. �5
15. �7 16. �15
17. �14 18. �9
Find x and y in each figure.
19. 20.
Find m�1 in each figure.
21. 22.140�
32�
160�
1
20�
7x�
(8x � 10)�
(6y � 20)�
5x�
40�(3y � 1)�
x
w
z
y
1 2
43
65
8
9 1011 12
13 1415 16
7
sm
ut
1 2
4 3
658
91011
127
q
r
s
1 243
6587
© Glencoe/McGraw-Hill 134 Glencoe Geometry
In the figure, m�2 � 92 and m�12 � 74. Find the measure of each angle.
1. �10 2. �8
3. �9 4. �5
5. �11 6. �13
Find x and y in each figure.
7. 8.
Find m�1 in each figure.
9. 10.
11. PROOF Write a paragraph proof of Theorem 3.3.Given: � || m, m || nProve: �1 � �12
12. FENCING A diagonal brace strengthens the wire fence and prevents it from sagging. The brace makes a 50° angle with the wire as shown.Find y. 50�
y�
m
n
1 23 4
657 8
�
k
9 1011 12
144�
62�
1
100�
50�
1
3y� (2x � 13)�(5y � 4)�3x�
(9x � 12)�
(4y � 10)�
n
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rs
12
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65
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910
1112
1314
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7
Practice Angles and Parallel Lines
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
Reading to Learn MathematicsAngles and Parallel Lines
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
© Glencoe/McGraw-Hill 135 Glencoe Geometry
Less
on
3-2
Pre-Activity How can angles and lines be used in art?
Read the introduction to Lesson 3-2 at the top of page 133 in your textbook.• Your textbook shows a painting that contains two parallel lines and a
transversal. What is the name for �1 and �2?• What is the relationship between these two angles?
Reading the Lesson1. Choose the correct word to complete each sentence.
a. If two parallel lines are cut by a transversal, then alternate exterior angles are
(congruent/complementary/supplementary).
b. If two parallel lines are cut by a transversal, then corresponding angles are
(congruent/complementary/supplementary).
c. If parallel lines are cut by a transversal, then consecutive interior angles are
(congruent/complementary/supplementary).
d. In a plane, if a line is perpendicular to one of two parallel lines, then it is
(parallel/perpendicular/skew) to the other.
Use the figure for Exercises 2 and 3.
2. a. Name four pairs of vertical angles.
b. Name all angles that form a linear pair with �7.
c. Name all angles that are congruent to �1.
d. Name all angles that are congruent to �4.
e. Name all angles that are supplementary to �3.
f. Name all angles that are supplementary to �2.
3. Which conclusion(s) could you make about lines u and v if m�4 � m�1?
A. t || u B. t ⊥ u C. v ⊥ u D. v ⊥ t E. v || t
Helping You Remember4. How can you use an everyday meaning of the adjective alternate to help you remember
the types of angle pairs for two lines and a transversal?
t
u
v
12
34
65
78
© Glencoe/McGraw-Hill 136 Glencoe Geometry
More Optical IllusionsIn drawings, diagonal lines may create the illusion of depth.For example, the figure at the right can be thought of as picturing a flat figure or a cube. The optical illusions on this page involve depth perception.
Answer each question.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
1. How many cubes do you see in thedrawing?
3. Does the drawing show a view from the top or the bottom of the stairs?
2. Can this figure show an actual object?
4. Which line segment is longer, A�B� orC�D�? Measure to check your answer.
A
B C
D
5. Which person in the drawing at the right appears to be tallest? Measure to check your answer.
6. Draw two more objects the same size on the figure at the right. Does one appear larger than the other?
Study Guide and InterventionSlopes of Lines
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
© Glencoe/McGraw-Hill 137 Glencoe Geometry
Less
on
3-3
Slope of a Line The slope m of a line containing two points with coordinates (x1, y1)
and (x2, y2) is given by the formula m � �yx
2
2
�
�
yx
1
1�, where x1 � x2.
Find the slope of each line.For line p, let (x1, y1) be (1, 2) and (x2, y2) be (�2, �2).
m � �yx
2
2
�
�
yx
1
1�
� ���
22
��
21� or �
43�
For line q, let (x1, y1) be (2, 0) and (x2, y2) be (�3, 2).
m � �yx
2
2
�
�
yx
1
1�
� ��23��
02� or ��
25�
Determine the slope of the line that contains the given points.
1. J(0, 0), K(�2, 8) 2. R(�2, �3), S(3, �5)
3. L(1, �2), N(�6, 3) 4. P(�1, 2), Q(�9, 6)
5. T(1, �2), U(6, �2) 6. V(�2, 10), W(�4, �3)
Find the slope of each line.
7. AB��� 8. CD���
9. EM��� 10. AE���
11. EH��� 12. BM���
x
y
O
C(–2, 2)
A(–2, –2)
H(–1, –4)
B(0, 4)
M(4, 2)
E(4, –2)
D(0, –2)
x
y
O
(1, 2)
(–2, –2)
(–3, 2)
(2, 0)
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 138 Glencoe Geometry
Parallel and Perpendicular Lines If you examine the slopes of pairs of parallel linesand the slopes of pairs of perpendicular lines, where neither line in each pair is vertical, youwill discover the following properties.
Two lines have the same slope if and only if they are parallel.Two lines are perpendicular if and only if the product of their slopes is �1.
Study Guide and Intervention (continued)
Slopes of Lines
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
Find the slope of a lineparallel to the line containing A(�3, 4)and B(2, 5).Find the slope of AB���. Use (�3, 4) for (x1, y1)and use (2, 5) for (x2, y2).
m � �yx
2
2
�
�
yx
1
1�
� �2
5�
�
(�43)
� or �15�
The slope of any line parallel to AB��� must be �
15�.
Find the slope of a lineperpendicular to PQ��� for P(�2, �4) andQ(4, 3).Find the slope of PQ���. Use (�2, �4) for (x1, y1) and use (4, 3) for (x2, y2).
m � �yx
2
2
�
�
yx
1
1�
� �34
�
�
((�
�
42))
� or �76�
Since �76� � ���
67�� � �1, the slope of any line
perpendicular to PQ��� must be ��67�.
Example 1Example 1 Example 2Example 2
ExercisesExercises
Determine whether MN��� and RS��� are parallel, perpendicular, or neither.
1. M(0, 3), N(2, 4), R(2, 1), S(8, 4) 2. M(�1, 3), N(0, 5), R(2, 1), S(6, �1)
3. M(�1, 3), N(4, 4), R(3, 1), S(�2, 2) 4. M(0, �3), N(�2, �7), R(2, 1), S(0, �3)
5. M(�2, 2), N(1, �3), R(�2, 1), S(3, 4) 6. M(0, 0), N(2, 4), R(2, 1), S(8, 4)
Find the slope of MN��� and the slope of any line perpendicular to MN���.
7. M(2, �4), N(�2, �1) 8. M(1, 3), N(�1, 5)
9. M(4, �2), N(5, 3) 10. M(2, �3), N(�4, 1)
Skills PracticeSlopes of Lines
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
© Glencoe/McGraw-Hill 139 Glencoe Geometry
Less
on
3-3
Determine the slope of the line that contains the given points.
1. S(�1, 2), W(0, 4) 2. G(�2, 5), H(1, �7)
3. C(0, 1), D(3, 3) 4. J(�5, �2), K(5, �4)
Find the slope of each line.
5. NP��� 6. TW���
7. a line parallel to TW��� 8. a line perpendicular to NP���
Determine whether AB��� and MN��� are parallel, perpendicular, or neither.
9. A(0, 3), B(5, �7), M(�6, 7), N(�2, �1) 10. A(�1, 4), B(2, �5), M(�3, 2), N(3, 0)
11. A(�2, �7), B(4, 2), M(�2, 0), N(2, 6) 12. A(�4, �8), B(4, �6), M(�3, 5), N(�1, �3)
Graph the line that satisfies each condition.
13. slope � 3, contains A(0, 1) 14. slope � ��32�, contains R(�4, 5)
15. contains Y(3, 0), parallel to DJ��� 16. contains T(0, �2), perpendicular to CX���
with D(�3, 1) and J(3, 3) with C(0, 3) and X(2, �1)
T(0, –2)
C(0, 3)
X(2, –1)x
y
OD(–3, 1)
J(3, 3)
Y(3, 0)x
y
O
R(–4, 5)
x
y
O
A(0, 1)x
y
O
x
y
ON T
W
P
© Glencoe/McGraw-Hill 140 Glencoe Geometry
Determine the slope of the line that contains the given points.
1. B(�4, 4), R(0, 2) 2. I(�2, �9), P(2, 4)
Find the slope of each line.
3. LM��� 4. GR���
5. a line parallel to GR��� 6. a line perpendicular to PS���
Determine whether KM��� and ST��� are parallel, perpendicular, or neither.
7. K(�1, �8), M(1, 6), S(�2, �6), T(2, 10) 8. K(�5, �2), M(5, 4), S(�3, 6), T(3, �4)
9. K(�4, 10), M(2, �8), S(1, 2), T(4, �7) 10. K(�3, �7), M(3, �3), S(0, 4), T(6, �5)
Graph the line that satisfies each condition.
11. slope � ��12�, contains U(2, �2) 12. slope � �
43�, contains P(�3, �3)
13. contains B(�4, 2), parallel to FG��� 14. contains Z(�3, 0), perpendicular to EK���
with F(0, �3) and G(4, �2) with E(�2, 4) and K(2, �2)
15. PROFITS After Take Two began renting DVDs at their video store, business soared.Between 2000 and 2003, profits increased at an average rate of $12,000 per year. Totalprofits in 2003 were $46,000. If profits continue to increase at the same rate, what willthe total profit be in 2009?
K(2, –2)
E(–2, 4)
Z(–3, 0)
x
y
O
F(0, –3)
B(–4, 2)
G(4, –2)x
y
O
P(–3, –3)
x
y
OU(2, –2)
x
y
O
x
y
O
R
G
S
P
M
L
Practice Slopes of Lines
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
Reading to Learn MathematicsSlopes of Lines
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
© Glencoe/McGraw-Hill 141 Glencoe Geometry
Less
on
3-3
Pre-Activity How is slope used in transportation?
Read the introduction to Lesson 3-3 at the top of page 139 in your textbook.• If you are driving uphill on a road with a 4% grade, how many feet will
the road rise for every 1000 horizontal feet traveled? • If you are driving downhill on a road with a 7% grade, how many meters
will the road fall for every 500 meters traveled?
Reading the Lesson1. Which expressions can be used to represent the slope of the line containing points (x1, y1)
and (x2, y2)? Assume that no denominator is zero.
A. ��
�
yx� B. �hv
oreirztoicnatlarlirsuen
� C. �yx
2
2
�
�
yx
1
1� D. �
cchh
aann
ggee
iinn
xy�
E. �yx
2
1
�
�
yx
1
2� F. �
yx
1
1
�
�
yx
2
2� G. �
xy
2
2
�
�
xy
1
1� H. �
yy
2
1
�
�
xx
2
1�
2. Match the description of a line from the first column with the description of its slopefrom the second column.
Type of Line Slope
a. a horizontal line i. a negative number
b. a line that rises from left to right ii. 0
c. a vertical line iii. undefined
d. a line that falls from left to right iv. a positive number
3. Find the slope of each line.
a. a line parallel to a line with slope �34�
b. a line perpendicular to the x-axis
c. a line perpendicular to a line with slope 5
d. a line parallel to the x-axis
e. y-axis
Helping You Remember
4. A good way to remember something is to explain it to someone else. Suppose your friendthinks that perpendicular lines (if neither line is vertical) have slopes that arereciprocals of each other. How could you explain to your friend that this is incorrect andgive her a good way to remember the correct relationship?
© Glencoe/McGraw-Hill 142 Glencoe Geometry
Slopes and PolygonsIn coordinate geometry, the slopes of two lines determine if the lines areparallel or perpendicular. This knowledge can be useful when working withpolygons.
1. The coordinates of the vertices of a triangle are A(�6, 4), B(8, 6), and C(4, �4). Graph �ABC.
2. J, K, and L are midpoints of A�B�, B�C�, and A�C�,respectively. Find the coordinates of J, K, and L.Draw �JKL.
3. Which segments appear to be parallel?
4. Show that the segments named in Exercise 3 are parallel by finding the slopes of all six segments.
The coordinates of the vertices of right �PQR are given. Find the slope of eachside of the triangle. Then name the hypotenuse.
5. P(5, 1) Q(1, �1) R(�2, 5) 6. P(�2, �3) Q(5, 1) R(2, 3)
slope of P�Q� � slope of P�Q� �
slope of Q�R� � slope of Q�R� �
slope of P�R� � slope of P�R� �
hypotenuse: hypotenuse:
The coordinates of quadrilateral PQRS are given. Graph quadrilateral PQRS andfind the slopes of the diagonals. State whether the diagonals are perpendicular.
7. P(�2, 6), Q(4, 0), R(1, �4), S(�5, 2) 8. P(0, 6), Q(3, 0), R(�4, �2), S(�5, 4)
S
P
Q
R
x
y
O
S
P
Q
R
x
y
O
A
B
C
K
J
L x
y
O
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
Study Guide and InterventionEquations of Lines
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
© Glencoe/McGraw-Hill 143 Glencoe Geometry
Less
on
3-4
Write Equations of Lines You can write an equation of a line if you are given any ofthe following:• the slope and the y-intercept,• the slope and the coordinates of a point on the line, or• the coordinates of two points on the line.
If m is the slope of a line, b is its y-intercept, and (x1, y1) is a point on the line, then:• the slope-intercept form of the equation is y � mx � b,• the point-slope form of the equation is y � y1 � m(x � x1).
Write an equation inslope-intercept form of the line withslope �2 and y-intercept 4.y � mx � b Slope-intercept form
y � �2x � 4 m � �2, b � 4
The slope-intercept form of the equation ofthe line is y � �2x � 4.
Write an equation inpoint-slope form of the line with slope ��
34� that contains (8, 1).
y � y1 � m(x � x1) Point-slope form
y � 1 � ��34�(x � 8) m � ��
34
�, (x1, y1) � (8, 1)
The point-slope form of the equation of theline is y � 1 � ��
34�(x � 8).
Example 1Example 1 Example 2Example 2
ExercisesExercises
Write an equation in slope-intercept form of the line having the given slope and y-intercept.
1. m: 2, y-intercept: �3 2. m: ��12�, y-intercept: 4
3. m: �14�, y-intercept: 5 4. m: 0, y-intercept: �2
5. m: ��53�, y-intercept: �
13� 6. m: �3, y-intercept: �8
Write an equation in point-slope form of the line having the given slope thatcontains the given point.
7. m � �12�, (3, �1) 8. m � �2, (4, �2)
9. m � �1, (�1, 3) 10. m � �14�, (�3, �2)
11. m � ��52�, (0, �3) 12. m � 0, (�2, 5)
© Glencoe/McGraw-Hill 144 Glencoe Geometry
Write Equations to Solve Problems Many real-world situations can be modeledusing linear equations.
Donna offers computer services to small companies in her city. Shecharges $55 per month for maintaining a web site and $45 per hour for eachservice call.
Study Guide and Intervention (continued)
Equations of Lines
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
ExampleExample
a. Write an equation torepresent the totalmonthly cost C formaintaining a web siteand for h hours ofservice calls.
For each hour, the costincreases $45. So the rate ofchange, or slope, is 45. They-intercept is located wherethere are 0 hours, or $55.C � mh � b
� 45h � 55
b. Donna may change her costs to represent themby the equation C � 25h � 125, where $125 is thefixed monthly fee for a web site and the cost perhour is $25. Compare her new plan to the old one if a company has 5 �
12� hours of service calls. Under
which plan would Donna earn more?
First plan
For 5�12� hours of service Donna would earn
C � 45h � 55 � 45�5�12�� � 55
� 247.5 � 55 or $302.50
Second Plan
For 5�12� hours of service Donna would earn
C � 25h � 125 � 25(5.5) � 125
� 137.5 � 125 or $262.50Donna would earn more with the first plan.
ExercisesExercises
For Exercises 1–4, use the following information.Jerri’s current satellite television service charges a flat rate of $34.95 per month for the basicchannels and an additional $10 per month for each premium channel. A competing satellitetelevision service charges a flat rate of $39.99 per month for the basic channels and anadditional $8 per month for each premium channel.
1. Write an equation in slope-intercept formthat models the total monthly cost foreach satellite service, where p is thenumber of premium channels.
3. A third satellite company charges a flatrate of $69 for all channels, including thepremium channels. If Jerri wants to adda fourth premium channel, which servicewould be least expensive?
2. If Jerri wants to include three premiumchannels in her package, which servicewould be less, her current service or thecompeting service?
4. Write a description of how the fee for thenumber of premium channels is reflectedin the equation.
Skills PracticeEquations of Lines
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
© Glencoe/McGraw-Hill 145 Glencoe Geometry
Less
on
3-4
Write an equation in slope-intercept form of the line having the given slope and y-intercept.
1. m: �4, y-intercept: 3 2. m: 3, y-intercept: �8
3. m: �37�, (0, 1) 4. m: ��
25�, (0, �6)
Write equations in point-slope form and slope-intercept form of the line havingthe given slope and containing the given point.
5. m: 2, (5, 2) 6. m: �3, (2, �4)
7. m: ��12�, (�2, 5) 8. m: �
13�, (�3, �8)
Write an equation in slope-intercept form for each line.
9. r 10. s
11. t 12. u
13. the line parallel to line r that contains (1, �1)
14. the line perpendicular to line s that contains (0, 0)
Write an equation in slope-intercept form for the line that satisfies the givenconditions.
15. m � 6, y-intercept � �2 16. m � ��53�, y-intercept � 0
17. m � �1, contains (0, �6) 18. m � 4, contains (2, 5)
19. contains (2, 0 ) and (0, 10) 20. x-intercept is �2, y-intercept is �1
x
y
O
r
s
t
u
© Glencoe/McGraw-Hill 146 Glencoe Geometry
Write an equation in slope-intercept form of the line having the given slope and y-intercept.
1. m: �23�, y-intercept: �10 2. m: ��
79�, �0, ��
12�� 3. m: 4.5, (0, 0.25)
Write equations in point-slope form and slope-intercept form of the line havingthe given slope and containing the given point.
4. m: �32�, (4, 6) 5. m: ��
65�, (�5, �2)
6. m: 0.5, (7, �3) 7. m: �1.3, (�4, 4)
Write an equation in slope-intercept form for each line.
8. b 9. c
10. parallel to line b, contains (3, �2)
11. perpendicular to line c, contains (�2, �4)
Write an equation in slope-intercept form for the line that satisfies the givenconditions.
12. m � ��49�, y-intercept � 2 13. m � 3, contains (2, �3)
14. x-intercept is �6, y-intercept is 2 15. x-intercept is 2, y-intercept is �5
16. passes through (2, �4) and (5, 8) 17. contains (�4, 2) and (8, �1)
18. COMMUNITY EDUCATION A local community center offers self-defense classes forteens. A $25 enrollment fee covers supplies and materials and open classes cost $10each. Write an equation to represent the total cost of x self-defense classes at thecommunity center.
x
y
O
c
b
Practice Equations of Lines
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
Reading to Learn MathematicsEquations of Lines
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
© Glencoe/McGraw-Hill 147 Glencoe Geometry
Less
on
3-4
Pre-Activity How can the equation of a line describe the cost of cellulartelephone service?
Read the introduction to Lesson 3-4 at the top of page 145 in your textbook.If the rates for your cellular phone plan are described by the equation inyour textbook, what will be the total charge (excluding taxes and fees) for amonth in which you use 50 minutes of air time?
Reading the Lesson1. Identify what each formula represents.
a. y � y1 � m(x � x1)
b. m � �yx
2
2
�
�
yx
1
1�
c. y � mx � b
2. Write the point-slope form of the equation for each line.
a. line with slope ��12� containing (�2, 5)
b. line containing (�4.5, �6.5) and parallel to a line with slope 0.5
3. Which one of the following correctly describes the y-intercept of a line?
A. the y-coordinate of the point where the line intersects the x-axis
B. the x-coordinate of the point where the line intersects the y-axis
C. the y-coordinate of the point where the line crosses the y-axis
D. the x-coordinate of the point where the line crosses the x-axis
E. the ratio of the change in y-coordinates to the change in x-coordinates
4. Find the slope and y-intercept of each line.
a. y � 2x � 7
b. x � y � 8.5
c. 2.4x � y � 4.8
d. y � 7 � x � 12
e. y � 5 � �2(x � 6)
Helping You Remember5. A good way to remember something new is to relate it to something you already know.
How can the slope formula help you to remember the equation for the point-slope form of a line?
© Glencoe/McGraw-Hill 148 Glencoe Geometry
Absolute ZeroAll matter is made up of atoms and molecules that are in constantmotion. Temperature is one measure of this motion. Absolute zerois the theoretical temperature limit at which the motion of themolecules and atoms of a substance is the least possible.
Experiments with gaseous substances yield data that allow you toestimate just how cold absolute zero is. For any gas of a constantvolume, the pressure, expressed in a unit called atmospheres,varies linearly as the temperature. That is, the pressure P and thetemperature t are related by an equation of the form P � mt � b,where m and b are real numbers.
1. Sketch a graph for the data in the table.
2. Use the data and your graph to find values for m and b in the equationP � mt � b, which relates temperature to pressure.
3. Estimate absolute zero in degrees Celsius by setting P equal to 0 in the equation above and using the values m and b that you obtained in Exercise 2.
25–25 50 75 100 125
0.5
1.5
1.0
t
P
O
t P(in �C) (in atmospheres)
�25 0.91
0 1.00
25 1.09
100 1.36
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
Study Guide and InterventionProving Lines Parallel
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
© Glencoe/McGraw-Hill 149 Glencoe Geometry
Less
on
3-5
Identify Parallel Lines If two lines in a plane are cut by a transversal and certainconditions are met, then the lines must be parallel.
If then
• corresponding angles are congruent, the lines are parallel.• alternate exterior angles are congruent,• consecutive interior angles are supplementary,• alternate interior angles are congruent, or• two lines are perpendicular to the same line,
If m�1 � m�2,determine which lines, if any, areparallel.
Since m�1 � m�2, then �1 � �2. �1 and�2 are congruent corresponding angles, so r || s.
n
m
r s
1 2
Find x and m�ABCso that m || n .
We can conclude that m || n if alternateinterior angles are congruent.
m�DAB � m�CDA3x � 10 � 6x � 20
10 � 3x � 2030 � 3x10 � x
m�ABC � 6x � 20� 6(10) � 20 or 40
n
mA
B
C
D
(3x � 10)�
(6x � 20)�
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find x so that � || m .
1. 2. 3.
4. 5. 6.m
n
�
(5x � 20)�
70�
m�
(3x � 20)�2x�
m�
(9x � 1)�(8x � 8)�
m�
(3x � 15)�
m� (4x � 20)�
6x �m�
(5x � 5)�
(6x � 20)�
© Glencoe/McGraw-Hill 150 Glencoe Geometry
Prove Lines Parallel You can prove that lines are parallel by using postulates andtheorems about pairs of angles. You also can use slopes of lines to prove that two lines areparallel or perpendicular.
Study Guide and Intervention (continued)
Proving Lines Parallel
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
ExampleExample
For Exercises 1–6, fill in the blanks.Given: �1 � �5, �15 � �5Prove: � || m, r || s
Statements Reasons
1. �15 � �5 1.
2. �13 � �15 2.
3. �5 � �13 3.
4. r || s 4.
5. 5. Given
6. 6. If corr � are �, then lines ||.
7. Determine whether PQ��� ⊥ TQ���. Explain why or why not.
x
y
O
P Q
T
�
m
r s
1 24 3
5 68 7
9 101112
14131516
a Given: �1 � �2, �1 � �3
Prove: A�B� || D�C�
Statements Reasons1. �1 � �2 1. Given
�1 � �32. �2 � �3 2. Transitive Property of �3. A�B� || D�C� 3. If alt. int. angles are �, then
the lines are ||.
1 2
3A B
CD
b. Which lines are parallel?Which lines are perpendicular?
slope of P�Q� � 0 slope of S�R� � 0
slope of P�S� � �43� slope of Q�R� � �
43�
slope of P�R� � �2 slope of S�Q� � �12�
So P�Q� || S�R�, P�S� || Q�R�, and P�R� ⊥ S�Q�.
x
y
O 4 8–4–8
8
4
–4
–8
P(–2, 4) Q(8, 4)
S(–8, –4) R(2, –4)
ExercisesExercises
Skills PracticeProving Lines Parallel
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
© Glencoe/McGraw-Hill 151 Glencoe Geometry
Less
on
3-5
Given the following information, determine which lines,if any, are parallel. State the postulate or theorem that justifies your answer.
1. �3 � �7 2. �9 � �11
3. �2 � �16 4. m�5 � m�12 � 180
Find x so that � || m .
5. 6. 7.
8. PROOF Provide a reason for each statement in the proof of Theorem 3.7.Given:�1 and �2 are complementary.
B�C� ⊥ C�D�Prove: B�A� || C�D�Proof:Statements Reasons1. B�C� ⊥ C�D� 1.2. m�ABC � m�1 � m�2 2.3. �1 and �2 are complementary. 3.4. m�1 � m�2 � 90 4.5. m�ABC � 90 5.6. B�A� ⊥ B�C� 6.7. B�A� || C�D� 7.
Determine whether each pair of lines is parallel. Explain why or why not.
9. 10.
x
y
O
U(4, 2)
T(–4, 0)
S(5, –3)R(0, –4)
x
y
O
A(1, 3)
B(–2, –3)E(0, –3)
F(2, 1)
1
A D
CB 2
m
k�(6x � 4)�
(8x � 8)�mk
�
(4x � 10)�
(3x � 10)�m
k �
(2x � 6)�
130�
m
�
a b1 2 3 4
8 7 6 5
9 10 11 1216 15 14 13
© Glencoe/McGraw-Hill 152 Glencoe Geometry
Given the following information, determine which lines,if any, are parallel. State the postulate or theorem that justifies your answer.
1. m�BCG � m�FGC � 180 2. �CBF � �GFH
3. �EFB � �FBC 4. �ACD � �KBF
Find x so that � || m .
5. 6. 7.
8. PROOF Write a two-column proof.Given:�2 and �3 are supplementary.Prove: A�B� || C�D�
9. LANDSCAPING The head gardener at a botanical garden wants to plant rosebushes inparallel rows on either side of an existing footpath. How can the gardener ensure thatthe rows are parallel?
1 42 5
3 6AB
CD
(2x � 12)�(5x � 15)�
t
m
�
(5x � 18)�(7x � 24)�
t
m
�
(3x � 6)�
(4x � 6)�
tm
�
A
B CD
E FH
K
GJ
Practice Proving Lines Parallel
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
Reading to Learn MathematicsProving Lines Parallel
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
© Glencoe/McGraw-Hill 153 Glencoe Geometry
Less
on
3-5
Pre-Activity How do you know that the sides of a parking space are parallel?
Read the introduction to Lesson 3-5 at the top of page 151 in your textbook.
How can the workers who are striping the parking spaces in a parking lotcheck to see if the sides of the spaces are parallel?
Reading the Lesson
1. Choose the word or phrase that best completes each sentence.
a. If two coplanar lines are cut by a transversal so that corresponding angles are
congruent, then the lines are (parallel/perpendicular/skew).
b. In a plane, if two lines are perpendicular to the same line, then they are
(perpendicular/parallel/skew).
c. For a line and a point not on the line, there exists
(at least one/exactly one/at most one) line through the point that is parallel to thegiven line.
d. If two coplanar lines are cut by a transversal so that consecutive interior angles are
(complementary/supplementary/congruent), then the lines are
parallel.
e. If two coplanar lines are cut by a transversal so that alternate interior angles are
congruent, then the lines are (perpendicular/parallel/skew).
2. Which of the following conditions verify that p || q ?
A. �6 � �12 B. �2 � �4
C. �8 � �16 D. �11 � �13
E. �6 and �7 are supplementary. F. �1 � �15
G. �7 and �10 are supplementary. H. �4 � �16
Helping You Remember
3. A good way to remember something new is to draw a picture. How can a sketch help youto remember the Parallel Postulate?
p
t
q
r
1 2
910
1213
14
1516
11
3 46 5
78
© Glencoe/McGraw-Hill 154 Glencoe Geometry
Scrambled-Up Proof
The reasons necessary to complete the following proof arescrambled up below. To complete the proof, number thereasons to match the corresponding statements.
Given: C�D� ⊥ B�E�A�B� ⊥ B�E�A�D� � C�E�B�D� � D�E�
Prove: A�D� || C�E�
Proof:
Statements Reasons
1. C�D� ⊥ B�E� Definition of Right Triangle
2. AB ⊥ B�E� Given
3. �3 and �4 are right angles. Given
4. �ABD and �CDE are right triangles. Definition of Perpendicular Lines
5. A�D� � C�E� Given
6. B�D� � D�E� CPCTC
7. �ABD � �CDE In a plane, if two lines are cut by a transversalso that a pair of corresponding angles iscongruent, then the lines are parallel.(Theorem 7-5)
8. �1 � �2 Given
9. A�D� || C�E� HL
A C
B ED
5
3 1 4 2
6
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
Study Guide and InterventionPerpendiculars and Distance
NAME ______________________________________________ DATE ____________ PERIOD _____
3-63-6
© Glencoe/McGraw-Hill 155 Glencoe Geometry
Less
on
3-6
Distance From a Point to a Line When a point is not on a line, the distance from the point to the line is the length of the segment that contains the point and isperpendicular to the line.
Draw the segment that represents the distance
from E to AF���.Extend AF���. Draw E�G� ⊥ AF���.E�G� represents the distance from E to AF���.
Draw the segment that represents the distance indicated.
1. C to AB��� 2. D to AB���
3. T to RS��� 4. S to PQ���
5. S to QR��� 6. S to RT���
XR T
S
R
T
S
P
Q
X
RT
S
P QX
U
RS
T
X
A
CD
BXA
C
BX
A F G
B E
A F
B E
Pdistance betweenM and PQ���
Q
M
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 156 Glencoe Geometry
Distance Between Parallel Lines The distance between parallel lines is the lengthof a segment that has an endpoint on each line and is perpendicular to them. Parallel linesare everywhere equidistant, which means that all such perpendicular segments have thesame length.
Find the distance between the parallel lines � and m whoseequations are y � 2x � 1 and y � 2x � 4, respectively.
Study Guide and Intervention (continued)
Perpendiculars and Distance
NAME ______________________________________________ DATE ____________ PERIOD _____
3-63-6
ExampleExample
Draw a line p through (0, 1) that isperpendicular to � and m .
Line p has slope ��12� and y-intercept 1. An
equation of p is y � ��12�x � 1. The point of
intersection for p and � is (0, 1).
x
y
O
mp �
(0, 1)
x
y
O
m� To find the point of intersection of p and m ,solve a system of equations.
Line m : y � 2x � 4Line p: y � ��
12�x � 1
Use substitution.
2x � 4 � ��12�x � 1
4x � 8 � �x � 25x � 10x � 2
Substitute 2 for x to find the y-coordinate.
y � ��12�x � 1
� ��12�(2) � 1 � �1 � 1 � 0
The point of intersection of p and m is (2, 0).
Use the Distance Formula to find thedistance between (0, 1) and (2, 0).
d � �(x2 ��x1)2 �� ( y2 �� y1)2�
� �(2 � 0�)2 � (�0 � 1�)2�� �5�
The distance between � and m is �5� units.
ExercisesExercises
Find the distance between each pair of parallel lines.
1. y � 8 2. y � x � 3 3. y � �2xy � �3 y � x � 1 y � �2x � 5
Skills PracticePerpendiculars and Distance
NAME ______________________________________________ DATE ____________ PERIOD _____
3-63-6
© Glencoe/McGraw-Hill 157 Glencoe Geometry
Less
on
3-6
Draw the segment that represents the distance indicated.
1. B to AC��� 2. G to EF��� 3. Q to SR���
Construct a line perpendicular to � through K. Then find the distance from K to �.
4. 5.
Find the distance between each pair of parallel lines.
6. y � 7 7. x � �6 8. y � 3xy � �1 x � 5 y � 3x � 10
9. y � �5x 10. y � x � 9 11. y � �2x � 5y � �5x � 26 y � x � 3 y � �2x � 5
x
y
O
�
K
x
y
O
�
K
S
P Q
RD
E F
GA
B
C
© Glencoe/McGraw-Hill 158 Glencoe Geometry
Draw the segment that represents the distance indicated.
1. O to MN��� 2. A to DC��� 3. T to VU���
Construct a line perpendicular to � through B. Then find the distance from B to �.
4. 5.
Find the distance between each pair of parallel lines.
6. y � �x 7. y � 2x � 7 8. y � 3x � 12y � �x � 4 y � 2x � 3 y � 3x � 18
9. Graph the line y � �x � 1. Construct a perpendicular segment through the point at (�2, �3). Then find the distance from the point to the line.
10. CANOEING Bronson and a friend are going to carry a canoe across a flat field to thebank of a straight canal. Describe the shortest path they can use.
(–2, –3)
y � �x � 1
x
y
O
x
y
O
�
B
x
y
O
�
B
T
VW
S U
A B
CD
M N
O
Practice Perpendiculars and Distance
NAME ______________________________________________ DATE ____________ PERIOD _____
3-63-6
Reading to Learn MathematicsPerpendiculars and Distance
NAME ______________________________________________ DATE ____________ PERIOD _____
3-63-6
© Glencoe/McGraw-Hill 159 Glencoe Geometry
Less
on
3-6
Pre-Activity How does the distance between parallel lines relate to hanging new shelves?
Read the introduction to Lesson 3-6 at the top of page 159 in your textbook.
Name three examples of situations in home construction where it would beimportant to construct parallel lines.
Reading the Lesson1. Fill in the blank with a word or phrase to complete each sentence.
a. The distance from a line to a point not on the line is the length of the segment
to the line from the point.
b. Two coplanar lines are parallel if they are everywhere .
c. In a plane, if two lines are both equidistant from a third line, then the two lines are
to each other.
d. The distance between two parallel lines measured along a perpendicular to the two
lines is always .
e. To measure the distance between two parallel lines, measure the distance between
one of the lines and any point on the .
2. On each figure, draw the segment that represents the distance indicated.
a. P to � b. D to AB���
c. E to FG��� d. U to RV���
Helping You Remember3. A good way to remember a new word is to relate it to words that use the same root. Use
your dictionary to find the meaning of the Latin root aequus. List three words other thanequal and equidistant that are derived from this root and give the meaning of each.
TV
R S
U
E
F G
A
D C
B
P
�
© Glencoe/McGraw-Hill 160 Glencoe Geometry
Parallelism in SpaceIn space geometry, the concept of parallelism must be extended to include two planes and a line and a plane.
Definition: Two planes are parallel if and only if they do not intersect.
Definition: A line and a plane are parallel if and only if they do not intersect.
Thus, in space, two lines can be intersecting, parallel, or skew while two planes or a line and a plane can only beintersecting or parallel. In the figure at the right, t � M,t � P, P || H, and � and n are skew.
The following five statements are theorems about parallel planes.
Theorem: Two planes perpendicular to the same line are parallel.Theorem: Two planes parallel to the same plane are parallel.Theorem: A line perpendicular to one of two parallel planes is
perpendicular to the other.Theorem: A plane perpendicular to one of two parallel planes is
perpendicular to the other.Theorem: If two parallel planes each intersect a third plane, then
the two lines of intersection are parallel.
Use the figure given above for Exercises 1–10. State yes or no to tell whether thestatement is true.
1. M � P 2. � � n 3. M � H 4. � � P 5. � � t
6. n � H 7. � � P 8. t � H 9. M � t 10. t � H
Make a small sketch to show that each statement is false.
11. If two lines are parallel to the same 12. If two planes are parallel, then any line plane, then the lines are parallel. in one plane is parallel to any line in the
other plane.
13. If two lines are parallel, then any plane 14. If two lines are parallel, then any plane containing one of the lines is parallel to containing one of the lines is parallel to any plane containing the other line. the other line.
t
� M
n P
H
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
3-63-6
Chapter 3 Test, Form 133
© Glencoe/McGraw-Hill 161 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of each question.
For Questions 1–3, refer to the figure at the right.
1. Identify the plane parallel to plane BCD.A. plane ABE B. plane ABFC. plane AEF D. plane DEF
2. Identify a segment parallel to C�D�.A. A�B� B. A�E� C. B�C� D. E�F�
3. Which segment is skew to D�E�?A. A�B� B. B�C� C. B�D� D. C�D�
For Questions 4–7, refer to the figure at the right.
Identify the special name for each angle pair.
4. �1 and �8A. alternate exterior B. alternate interiorC. consecutive interior D. corresponding
5. �3 and �7A. alternate exterior B. alternate interiorC. consecutive interior D. corresponding
6. Given a || b and m�2 � 65, find m�6.A. 25 B. 65 C. 115 D. 140
7. Given a || b and m�3 � 5x � 10 and m�5 � 3x � 10, find x.A. 110 B. 70 C. 20 D. 2.5
For Questions 8–10, refer to the figure at the right.
8. Which angle relationship justifies that � || m?A. �1 � �7 B. �3 � �4C. �4 � �5 D. �6 � �8
9. If m�2 � 6x � 8 and m�6 � 8x � 6, find x so that � || m .A. �7 B. 1 C. 7 D. 14
10. Given m�6 � m�7 � 180, which postulate or theorem justifies that � || m ?A. Consecutive Interior Angles TheoremB. Corresponding Angles PostulateC. Alternate Exterior Angles TheoremD. Alternate Interior Angles Theorem
m
�
p
12
34
65
78
b
a 1 23 4
657 8
B
A
DF
C
E
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 162 Glencoe Geometry
Chapter 3 Test, Form 1 (continued)33
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
For Questions 11–12, determine the slope of the line that contains thegiven points.
11. A(0, 5), B(5, 0)A. �1 B. 0 C. 1 D. 5
12. F(�2, �4), G(1, 2)
A. �2 B. ��12� C. �
12� D. 2
13. What is the slope of a line parallel to the line containing (�6, 1) and (3, �2)?
A. �3 B. � �13� C. �
13� D. 3
14. Find the slope of the line perpendicular to the line containing (0, 0) and (�1, 4).
A. � �14� B. �4 C. �
14� D. 4
15. Which is an equation of the line with slope 4 and a y-intercept �3?
A. y � �3x � 4 B. y � �3x � �34� C. y � 4x � 3 D. y � 4x � �
34�
16. Which is an equation of the line with slope 2 that contains (3, 1)?A. y � 1 � 2(x � 3) B. y � 1 � 2(x � 3)C. y � 3 � 2(x � 1) D. y � 3 � (x � 2)
17. Yoga lessons cost $5 per lesson if Kylie enrolls in the health club for a fee of $120 per year. Suppose Kylie joins the health club. Which equationrepresents the yearly cost C of � yoga lessons?A. C � 5� B. C � 5� � 120C. C � 5� � 120 D. C � 5(� � 120)
18. What is the distance from P to n shown in the figure?A. �3B. 1C. 4D. 5
For Questions 19–20, find the distance between each pair of parallel lines.
19. y � 4 and y � 6A. 2 B. 4 C. 6 D. 10
20. y � x and y � x � 2A. 1 B. 1.5 C. �2� D. 2
Bonus What is the slope of a line perpendicular to y � �2?
P
x
y
O
n
B:
NAME DATE PERIOD
Chapter 3 Test, Form 2A33
© Glencoe/McGraw-Hill 163 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of each question.
For Questions 1 and 2, refer to the figure at the right.
1. Identify the plane parallel to plane PQT.A. plane PQS B. plane PTSC. plane RSV D. plane TUW
2. Which segment is skew to R�V�?A. R�S� B. R�Q� C. S�W� D. S�P�
For Questions 3–10, refer to the figure at the right.
Identify the special name for each angle pair.
3. �3 and �10A. alternate exterior B. alternate interiorC. consecutive interior D. corresponding
4. �9 and �13A. alternate exterior B. alternate interiorC. consecutive interior D. corresponding
5. Given p || q and m�3 � 75, find m�5.A. 15 B. 75 C. 105 D. 120
6. Given p || q and m�10 � 3x � 7 and m�13 � 4x � 9, find x.A. �2 B. 2 C. 16 D. 28
7. Given �1 � �5, which postulate or theorem justifies that p || q?A. Corresponding Angles PostulateB. Consecutive Interior Angles TheoremC. Alternate Exterior Angles TheoremD. Alternate Interior Angles Theorem
8. If �12 � �14, which postulate or theorem justifies that p || q ?A. Corresponding Angles PostulateB. Consecutive Interior Angles TheoremC. Alternate Exterior Angles TheoremD. Alternate Interior Angles Theorem
9. If p || q by the Consecutive Interior Angles Theorem, which angle pair must be supplementary?A. �3 and �10 B. �3 and �8 C. �8 and �13 D. �15 and �16
10. If m�4 � 7x � 20 and m�8 � 5x � 18, find x so that p || q.A. �19 B. �1 C. 1 D. 19
r
s
p q
1 234 65
78
9 1011
13 14151612
V
R S
W
U T
Q P 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 164 Glencoe Geometry
Chapter 3 Test, Form 2A (continued)33
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
Determine the slope of the line that contains the given points.
11. P(�6, 3), Q(12, 9)
A. �3 B. ��13� C. �
13� D. 3
12. M(�8, 14), N(2, �11)
A. ��52� B. ��
25� C. �
25� D. �
52�
13. What is the slope of a line parallel to the line containing (�6, �6) and (9, 14)?
A. �34� B. �
43� C. �
130� D. undefined
14. Find the slope of a line perpendicular to the line containing (�8, 10) and (0, 9).
A. �8 B. ��18� C. �
18� D. 8
15. Which is an equation of the line with slope �12� that contains (�4, 7)?
A. y � 7 � �12�(x � 4) B. y � 7 � �
12�(x � 4)
C. y � 7 � �4x � �12� D. y � 7 � �
12�(x � 4)
16. Which is an equation of the line with x-intercept 2 and y-intercept 12?A. y � �6x � 12 B. y � 2x � 12 C. y � 6x � 12 D. y � 12x � 2
17. Which is an equation of the line containing (1, �3) and (7, 15)?A. y � �3x � 8 B. y � 3x C. y � 3x � 6 D. y � 3x � 10
18. Mr. Perugia gives 4 points per question for q questions on English quizzesplus 5 points for a bonus question. Which equation represents the total scoreT a student can receive on a quiz?A. T � 5 � 4q B. T � 4q � 5 C. T � 4(q � 5) D. 4T � q � 5
19. What is the distance from D to t shown in the figure?A. 2B. 3C. 5D. �5�
20. What is the distance between parallel lines whose equations are y � 2x � 7and y � 2x � 3?
A. �2� B. �5� C. 2�5� D. 4�2�
Bonus Suppose Ian reads at the rate of 15 pages an hour. Write an equation to represent the number of pages y Ian will still need to read after reading x hours in a 285-page novel. How long will it take Ian to read the entire novel?
D
x
y
O
t
B:
NAME DATE PERIOD
Chapter 3 Test, Form 2B33
© Glencoe/McGraw-Hill 165 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of each question.
For Questions 1 and 2, refer to the figure at the right.
1. Identify the plane parallel to plane ACE.A. plane ABG B. plane BCHC. plane EFK D. plane GJF
2. Which segment is skew to J�K�?A. A�B� B. C�D�C. J�H� D. G�F�
For Questions 3–10, refer to the figure at the right.
Identify the special name for each angle pair.
3. �1 and �5A. alternate exterior B. alternate interiorC. consecutive interior D. corresponding
4. �10 and �14A. alternate exterior B. alternate interiorC. consecutive interior D. corresponding
5. Given s || t and m�7 � 84, find m�10.A. 6 B. 84 C. 96 D. 104
6. Given s || t and m�1 � 8x � 4 and m�15 � 6x � 24, find x.A. �10 B. 14 C. 20 D. 28
7. Given �6 � �12, which postulate or theorem justifies that s || t?A. Corresponding Angles PostulateB. Consecutive Interior Angles TheoremC. Alternate Exterior Angles TheoremD. Alternate Interior Angles Theorem
8. If �8 � �16, which postulate or theorem justifies that s || t?A. Corresponding Angles PostulateB. Consecutive Interior Angles TheoremC. Alternate Exterior Angles TheoremD. Alternate Interior Angles Theorem
9. If s || t by the Alternate Exterior Angles Theorem, which angle pair must be congruent?A. �1 and �7 B. �1 and �5 C. �1 and �15 D. �1 and �13
10. If m�6 � 10x � 6 and m�11 � 4x � 18, find x so that s || t.A. 2 B. 4 C. 12 D. 32
a bs
t
1 2 3 46 578
9 10 111314151612
B
AE
F
K
J
C H
GD
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 166 Glencoe Geometry
Chapter 3 Test, Form 2B (continued)33
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
Determine the slope of the line that contains the given points.
11. C(1, �7), D(5, 13)
A. �5 B. � �15� C. �
15� D. 5
12. Y(�7, �4), Z(5, 2)
A. �1 B. �12� C. 1 D. 2
13. What is the slope of a line parallel to the line containing (2, 5) and (6, �11)?
A. �13 B. �4 C. ��14� D. 4
14. Find the slope of a line perpendicular to the line containing (�2, �9) and (8, 6).
A. � �23� B. �
25� C. �
23� D. �
32�
15. Which is an equation of the line with slope of ��35� that contains (5, �5)?
A. y � 5 � ��35�(x � 5) B. y � 5 � ��
35�(x � 5)
C. y � 5 � ��35�(x � 5) D. y � 5 � ��
35�x � 5
16. Which is an equation of the line with x-intercept 18 and a y-intercept 3?
A. y � ��16�x � 3 B. y � �
16�x � 3 C. y � 3x � 18 D. y � 6x � 3
17. Which is an equation of the line containing (�3, 13) and (6, �5)?A. y � 2x � 7 B. y � �2x � 12 C. y � �2x � 17 D. y � �2x � 7
18. Mrs. Ekeledo writes computer manuals. She charges $125 to review writingspecifications plus $50 per hour h to write the manual. Which equationrepresents the total fee F that Mrs. Ekeledo earns for writing each computermanual?A. F � 50(h � 125) B. F � 50h � 125C. F � 125 � 50h D. 50F � h � 125
19. What is the distance from R to q shown in the figure?
A. �3�B. 3
C. 3�2�D. �8�
20. What is the distance between parallel lines whose equations are y � �x � 2and y � �x � 8?A. 3�2� B. 2�17� C. 10 D. 2�29�
Bonus Suppose line a is perpendicular to line b and line b is parallel to line c. What is the relationship between line aand line c?
R
x
y
O
q
B:
NAME DATE PERIOD
Chapter 3 Test, Form 2C33
© Glencoe/McGraw-Hill 167 Glencoe Geometry
Ass
essm
ents
For Questions 1 and 2, refer to the figure.
1. Identify the intersection of plane SVX and plane STU.
2. Name a segment skew to W�Y�.
For Questions 3–5, identify each pair of angles as alternate interior, alternateexterior, corresponding, or consecutiveinterior angles.
3. �2 and �12
4. �3 and �5
5. �7 and �15
6. Given m || n and m�8 � 86, find m�13.
7. Find x and y given m || n, m�4 � 6x � 5, m�10 � 5x � 8, andm�9 � 3y � 10.
Determine the slope of the line that contains the givenpoints.
8. V(�10, �4), W(5, 5)
9. A(�2, 9), C(2, �15)
10. G(�6, 14), L(�3, 9)
For Questions 11–13, determine whether CS��� and KP��� areparallel, perpendicular, or neither.
11. C(1, �12), S(5, 4), K(1, 9), P(6, �6)
12. C(�5, 6), S(�3, 2), K(�2, 10), P(1, 4)
13. C(�6, �7), S(�3, �5), K(3, 3), P(9, 7)
14. Printer’s Ink charges $1.18 per page p to copy a report plus$12 to bind it. Write an equation that represents the total costC to copy and bind a report. What would be the cost to copyand bind a 50-page report?
m
n
s t
1 234
6578
9 1011
13 141516
12
V
S T
U
Z
X W
Y
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 168 Glencoe Geometry
Chapter 3 Test, Form 2C (continued)33
Write an equation in slope-intercept form for the line thatsatisfies the given conditions.
15. m � �9, y-intercept � 3
16. m � 3, contains (�1, 5)
17. x-intercept is 3, y-intercept is �1
18. contains (�7, 9) and (6, �4)
For Questions 19–21, given the following information, determine which lines, if any are parallel. State the postulate or theorem that justifies your answer.
19. �1 � �2
20. �DAB � �EBC
21. m�ADE � m�BED � 180
22. Find x so that a || b.
For Questions 23 and 24, find the distance between eachpair of parallel lines.
23. y � x � 6 and y � x � 8
24. y � �2x � 10 and y � �2x � 5
25. Construct a line perpendicular to � through B(�2, 5). Thenfind the distance from B to �.
Bonus Draw and label a figure you could use to prove thetheorem If two lines in a plane are cut by a transversal sothat a pair of consecutive interior angles is supplementary,then the lines are parallel. State the given and thestatement to be proved.
a
b
(9x � 44)�
(6x � 10)�
1
A
B
C F
E
D
2
NAME DATE PERIOD
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
B:
1
43
ms
�
B
x
y
O
�
Chapter 3 Test, Form 2D33
© Glencoe/McGraw-Hill 169 Glencoe Geometry
Ass
essm
ents
For Questions 1 and 2, refer to the figure.
1. Identify the intersection of plane ABD andplane CDE.
2. Name a segment skew to B�C�.
Identify each pair of angles as alternate interior, alternate exterior,corresponding, or consecutive interiorangles.
3. �5 and �13
4. �7 and �12
5. �9 and �16
6. Given a || b and m�6 � 89, find m�10.
7. Find x and y given a || b, m�8 � 4x � 10, m�12 � 7x � 17,and m�11 � 3y.
Determine the slope of the line that contains the givenpoints.
8. P(�5, 11), R(5, 7)
9. B(7, �1), G(14, 0)
10. U(�6, 9), V(�3, 8)
For Questions 11–13, determine whether BT��� and MV��� areparallel, perpendicular, or neither.
11. B(1, �4), T(5, 12), M(�8, 3), V(�4, 2)
12. B(�5, �7), T(10, 17), M(�5, �10), V(5, 6)
13. B(3, �5), T(5, �1), M(�2, 6), V(4, 3)
14. A job hotline service charges $25 plus a 4% commission c onthe first month of earnings if it finds work for a client. Writean equation that represents the total cost T to find a jobthrough the hotline. What is the cost if a client earns $1500the first month?
a b
d
e
12
34
56
78
910
1112
1314
1516
B A
DC
E
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 170 Glencoe Geometry
Chapter 3 Test, Form 2D (continued)33
Write an equation in slope-intercept form for the line thatsatisfies the given conditions.
15. m � 7, y-intercept � �8
16. m � �4, contains (�4, 8)
17. x-intercept is �5, y-intercept is 2
18. containing (2, 5) and (7, �5)
Given the following information,determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer.
19. �QSR � �SUT
20. �1 � �2
21. m�RTU � m�TUS � 180
22. Find x so that p || q.
Find the distance between each pair of parallel lines.
23. y � 3x � 1 and y � 3x � 11
24. y � �5x and y � �5x � 26
25. Construct a line perpendicular to � through Q(2, 3). Then findthe distance from Q to �.
Bonus Draw and label a figure you could use to prove the theoremIf two lines in a plane are cut by a transversal so that a pairof alternate interior angles is congruent, then the lines areparallel. State the given and the statement to be proved.
pq
(4x � 1)�
(12x � 11)�
1
P
R
TU
S
Q
2
NAME DATE PERIOD
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
B:
1
32
mt
�
x
y
O
�
Q
Chapter 3 Test, Form 333
© Glencoe/McGraw-Hill 171 Glencoe Geometry
Ass
essm
ents
1. Draw a triangular prism and label the parallel planes ABCand DEF.
2. Identify two planes in your triangular prism that intersect.Name their intersection.
3. Name two lines in your triangular prism that are skew.
Identify each pair of angles as alternate interior, alternate exterior, corresponding,or consecutive interior angles.
4. �9 and �12
5. �2 and �3
6. �4 and �11
7. If m�9 � 110 and m�8 � 30,find m�6.
8. Find x, y, and z in the figure.
Determine the slope of the line that contains the givenpoints.
9. D(�6, �7), F(12, 23)
10. V(�2, �0.25), U(4, �2.5)
Determine whether QV��� and RM��� are parallel,perpendicular, or neither.
11. Q(�3, �8), V(5, 12), R(�2.5, 1), M(�5, 2)
12. Q(�2, 4.5), V(4, 9), R(�4, �12), M(10, �1.5)
(8x � 7)�
(3x � 11)�(2y � 23)� (3z2 � 5)�
(4y � 8)�
p
q
rst
1 234
56
78
9 101112
1314
f
gh
12
34
56
7
89
10
11
12
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
B
C
F
D
A
E
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 172 Glencoe Geometry
Chapter 3 Test, Form 3 (continued)33
13. Write an equation of the line that is parallel to y � �85�x � 12
and contains (�6, �10).
14. Write an equation of the line that is perpendicular to 3x � 4y � 9 and contains (�4, �5).
15. A car rental company charges $30 per day plus $0.28 per mile mfor each mile over 100 miles to rent a car. Mr. Benitez rented acar for 5 days and drove 255 miles. Write an equation thatrepresents the total cost C to rent a car for 5 days. What wasMr. Benitez’s total cost?
Given the following information,determine which lines, if any areparallel. State the postulate or theorem that justifies your answer.
16. �RNS � �PSN
17. m�MRS � m�RSN � 180
18. Find x so that a || b.
19. Find the distance between parallel lines whose equations are
y � ��14�x � 2 and y � ��
14�x � �
94�.
20. Graph line m whose equation is �6x � 3y � 9. Construct aperpendicular line through P(3, 1). Then find the distancefrom P to m.
Bonus Suppose line p is perpendicular to line s and line q isperpendicular to line s. Are lines p and q parallel under allcircumstances? Draw a figure to illustrate your answer.
(6x � 13)�
(229 � 4x)�
ab
MN
P
S T
VU
R
NAME DATE PERIOD
13.
14.
15.
16.
17.
18.
19.
20.
B:
sp
q
x
y
OP
m
Chapter 3 Open-Ended Assessment33
© Glencoe/McGraw-Hill 173 Glencoe Geometry
Ass
essm
ents
Demonstrate your knowledge by giving a clear, concise solution toeach problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate beyond the requirements of the problem.
1. TOWN PLANNING The lines in the diagram represent the intersection of streets near Todd’s home. ●●A represents the location of Todd’s home, and●●B represents the location of a library 1, 2, 3, 4, and 5 represent the anglesformed at street intersections.
a. Suppose you want to determine whether the streets represented bylines a, b, and c are parallel. What information would you need? Explainyour reasoning.
b. Suppose the streets represented by lines d and e are parallel. If themeasure of �5 is 112, find the measure of �4. Explain how youdetermined the measure.
c. If m�1 � 3x � 7 and m�4 � 2x � 20, find x so that lines a and c areparallel. Explain how you arrived at your answer and describe whythese measures allow you to determine that a and c are parallel.
d. Todd wants to go from the library to his home. If the streets representedby lines a and b are parallel, how could Todd determine the shortestdistance between ●●B and street b on which he lives? Give an explanationand illustrate your answer on a sketch of the above diagram.
2. Draw a line � on a coordinate grid so that the line contains (�3, �1) and(2, 4).
a. Write an equation of the line in slope-intercept form. Provide anexplanation for each of the steps.
b. If you construct a line parallel to �, what is the slope of the line? Howdo you know? Construct a line parallel to � that contains (1, 5).
c. Find the distance between the two lines. Explain how you determinedthe distance.
1 Ba
b
cd e f
2 34 5
A
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 174 Glencoe Geometry
Chapter 3 Vocabulary Test/Review33
Write whether each statement is true or false. If false,replace the underlined word or number to make a truesentence.
For Questions 1–4, refer to the figure.
1. �4 and �5 are .
2. According to the , line ris parallel to line t given �3 � �8.
3. Given r || t, then �4 and �6 are supplementary.
4. Line p is a transversal since it intersects or more lines ina plane at different points.
5. When a linear equation is written in the form y � mx � b, mis the of the line and b is the y-intercept.
6. are located between the lines cut by a transversal.
7. If two lines do not intersect and are everywhere equidistant,the lines are .
8. The Perpendicular Transversal Theorem states that in aplane, if a line is perpendicular to one of two parallel lines,then it is to the other.
9. The equation y � 6 � ��58�(x � 2) is in .
10. The ratio of the rise to the run of a line is called its.
In your own words—
11. Describe what is meant by rate of change.
slope-intercept form
point-slope form
parallel
skew
Interior angles
slope
one
consecutive interior angles
Parallel Postulate
corresponding angles r
t
1 234
5687
p
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
alternate exterior anglesalternate interior anglesconsecutive interior anglescorresponding angles
equidistantnon-Euclidean geometryparallel linesparallel planes
plane Euclidean geometrypoint-slope formrate of changeskew lines
slopeslope-intercept formspherical geometrytransversal
NAME DATE PERIOD
SCORE
Chapter 3 Quiz (Lessons 3–1 and 3–2)
33
© Glencoe/McGraw-Hill 175 Glencoe Geometry
Ass
essm
ents
NAME DATE PERIOD
SCORE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
For Questions 1 and 2, refer to the figure.
1. Which plane is parallel to plane EGH?
2. Name the intersection of plane ABC and plane EFB.
For Questions 3–8, refer to the figure.
Identify each pair of angles as alternate interior, alternate exterior,corresponding, or consecutive interiorangles.
3. �2 and �10 4. �6 and �12
5. �1 and �5 6. �14 and �15
Given a || b and m�7 � 94, find the measure of thefollowing angles.
7. �10 8. �9
9. Find x and y in the figure.
10. STANDARDIZED TEST PRACTICEWhat is m�UVW?A. 39 B. 42C. 81 D. 138
138�V
U
W39�
(4y � 3)�(5x � 7)�
(3x � 17)�
a
b
1 2 3 4568
9 10 11 1213141516
7
dc
E
F
B A
K
J
HG
LMD
C
Chapter 3 Quiz (Lesson 3–3)
33
1.
2.
3.
4.
5.
Determine the slope of the line that contains the givenpoints.
1. E(�5, 6), Z(4, �3) 2. Y(�1, �12), P(3, 8)
3. B(�2, 7), N(8, �8) 4. R(�5, �7), Q(5, �5)
Determine whether DC��� and NM��� are parallel,perpendicular, or neither.
5. D(�4, �11), C(2, 7), N(�6, �1), M(3, �4)
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 176 Glencoe Geometry
Chapter 3 Quiz (Lessons 3–4 and 3–5)
33
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
1. Write an equation in point-slope form of the line with slope ��13�
that contains (3, 8).
2. Write an equation in slope-intercept form of the line with slope �
53� and y-intercept of �2.
3. Write an equation in slope-intercept form of the line thatcontains (�1, 7) and (3, �9).
4. A bottled water company charges $8 per month for a watercooler and $5 per bottle b of water. Write an equation thatrepresents the total cost C for monthly water service.
Given the following information,determine which lines, if any, are parallel. State the postulate or theoremthat justifies your answer.
5. �1 � �6 6. �2 � �3
7. �4 � �9 8. m�7 � m�6 � 180
9. Given g || h and m�8 � 64, find m�5.
10. If m�2 � 5x � 17 and m�7 � 3x � 35, find x so that p || q.
12
3 45
67 89
g
h
jp q
NAME DATE PERIOD
SCORE
Chapter 3 Quiz (Lesson 3–6)
33
1.
2.
3.
4.
1. Draw the segment that represents the distance from B to DC���.
2. Construct a line perpendicular to � through H(�1, 3). Thenfind the distance from H to �.
Find the distance between each pair of parallel lines.
3. y � �8 4. y � �x � 9y � 4 y � �x � 7
x
y
O
�H
BA
CD
NAME DATE PERIOD
SCORE
Chapter 3 Mid-Chapter Test (Lessons 3–1 through 3–3)
33
© Glencoe/McGraw-Hill 177 Glencoe Geometry
Ass
essm
ents
For Questions 1 and 2, refer to the figure.
1. Which segment is skew to I�J�?A. G�H� B. A�J�C. H�I� D. A�B�
2. Which plane is parallel to plane CDF?A. BEF B. HIJC. ABE D. ABC
For Questions 3–5, refer to the figure. Identify the special name for each angle pair.
3. �2 and �4A. alternate exterior B. alternate interiorC. corresponding D. consecutive interior
4. �3 and �12A. alternate exterior B. alternate interiorC. corresponding D. consecutive interior
5. Given p || r and m�8 � 119, find m�11.A. 29 B. 61 C. 119 D. 151
p
qr
s
12
34
56
8
910
1112
7
B A
EFI J
C
HG
D
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
1.
2.
3.
4.
5.
Part II
6. Find x and y in the figure.
For Questions 7 and 8, refer to the graph. Find the slope of each line.
7. k
8. m
Determine whether AB��� and DF��� are parallel, perpendicular,or neither.
9. A(�2, �11), B(6, 5), D(�7, �10), F(2, 8)
10. A(�1, 6), B(2, �9), D(�10, �1), F(5, 2)
x
y
O
k
m
(�2, 2)
(�4, �1) (0, �2)
(2, 4)
(5y � 11)�(4x � 6)�
(3x � 22)�
Part I Write the letter for the correct answer in the blank at the right of each question.
© Glencoe/McGraw-Hill 178 Glencoe Geometry
Chapter 3 Cumulative Review(Chapters 1–3)
33
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
1. Find y and BC if B is between A and C, AB � 6y, BC � 12y,and AB � 42. (Lesson 1-2)
2. What are the coordinates of D if E(�2, 2) is the midpoint ofD�F�, and F has coordinates (3, 5)? (Lesson 1-3)
For Questions 3 and 4, use the figure at the right. FA��� and FE��� are opposite rays.
3. Classify �AFB. (Lesson 1-4)
4. Find x and �BFC so that B�F� and F�D� areperpendicular. (Lesson 1-5)
5. Determine whether the conjecture is true or false. Give acounterexample for a false conjecture. (Lesson 2-1)
Given: � || mConjecture: The slopes of lines � and m are equal.
6. Identify the hypothesis and conclusion of the statementIf m�1 � m�2 � 180, then �1 and �2 are supplementary.(Lesson 2-3)
7. Determine whether statement (3) follows from statements (1)and (2) by the Law of Detachment or by the Law of Syllogism.Write invalid if neither law applies. (Lesson 2-4)
(1) If you swim, you must use earplugs.(2) Roy swims.(3) Roy must use earplugs.
8. Determine whether the statement If p || q and r ⊥ p, then r ⊥ qis always, sometimes, or never true. (Lesson 2-5)
9. If F(�1, 6), G(5, 3), J(2, �4), and K(6, �6), are FG��� and JK���
parallel, perpendicular, or neither? (Lesson 3-3)
10. Write an equation for a line in point-slope form with a slope of4 that contains the point at (2, 8). (Lesson 3-4)
11. Find the distance from A(�1, 5) to the line whose equation is4x � 5y � 12. (Lesson 3-6)
A
E
C
B
D
F(9x � 25)�
4x�
NAME DATE PERIOD
SCORE
Standardized Test Practice (Chapters 1–3)
© Glencoe/McGraw-Hill 179 Glencoe Geometry
For Questions 1–4, use the figure.
1. Points A, B, and D . (Lesson 1-1)
A. are collinear. B. are coplanar.C. lie on AD��� D. contain C.
2. Which segments are congruent? (Lesson 1-2)
E. A�B� and E�D� F. A�C� and E�D�G. A�C� and B�D� H. A�B� and B�D�
3. Find x. (Lesson 1-2)
A. 1 B. 1.5 C. 2 D. 2.7
4. Classify �C. (Lesson 1-4)
E. right angle F. acute angleG. obtuse angle H. straight angle
5. Which statement has the same truth value as 3 � 5? (Lesson 2-2)
A. 3 � x B. AB � 3C. AB � BC D. BC � 3 � x
6. The Law of Syllogism says that [(p → q) � (q → r)] → .(Lesson 2-4)
E. q F. r G. p → q H. p → r
7. State the property that justifies the statement If a � b � 25 and b � c, then a � c � 25. (Lesson 2-6)
A. Reflexive Property B. Symmetric PropertyC. Transitive Property D. Substitution Property
For Questions 8–10, use the figure.
8. What type of angles are �3 and �10? (Lesson 3-1)
E. alternate interior anglesF. alternate exterior anglesG. corresponding anglesH. consecutive interior angles
9. State the transversal that forms �11 and �13. (Lesson 3-1)
A. � B. m C. p D. q
10. If m�1 � 120, find m�8. (Lesson 3-2)
E. 60 F. 110 G. 120 H. 140
2 9 1012
131416
15
1143
mn p q
�
5 687
1
?
A B C
3 x
? 2.7 cm 2.7 cm
4.6 cm
3.5 cm (x � 2) cmC D
BA
E
NAME DATE PERIOD
SCORE 33
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10. E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
Ass
essm
ents
© Glencoe/McGraw-Hill 180 Glencoe Geometry
Standardized Test Practice (continued)
11. Find DE. (Lesson 1-2)
12. The measure of the complement of �A is 185less than two times the measure of thesupplement of �A. Find m�A. (Lesson 1-5)
13. The table below shows the number of each typeof question that will be on four quarterlyhistory tests. If Monique prefers the ratio ofmultiple choice questions to essay questions tobe 4:1, which test will she prefer? (Lesson 2-4)
TestNumber of Number of Multiple
Essay Questions Choice Questions
1 4 26
2 8 22
3 6 24
4 5 25
D F
60 � 4a
E
3a � 5 13
NAME DATE PERIOD
33
Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
Part 3: Short Response
Instructions: Show your work or explain in words how you found your answer.
11. 12.
13.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
14. Determine the number of line segments that can be drawn connecting this arrangement of points.(Lesson 2-5)
15. The distance between D(�2, �2) and E(a, �2) is 7. Find a.(Lesson 1-3)
16. The line that is perpendicular to y � �x and contains thepoint at (2, 4) is y � x � . (Lesson 3-4)
17. Find the slope of a line parallel to 3y � 6x � 9. (Lesson 3-4)
18. Find x so that � || m . (Lesson 3-5) mn
�(7x � 15)�
25�
?
14.
15.
16.
17.
18.
2 3 8 5
3
Unit 1 Review (Chapter 1–3)
33
© Glencoe/McGraw-Hill 181 Glencoe Geometry
Ass
essm
ents
1. Find JK. What is the precision for this measurement?
2. Find the distance between S(5, �7) and T(13, �9). Then findthe coordinates of the midpoint of S�T�.
3. In the figure, QP��� and QT��� are opposite rays. Find m�PQR, m�RQS, and m�SQT. Then classify each angle as right, acute, or obtuse.
4. Find the measures of two complementary angles if thedifference in the measures is 16.
5. Name the polygon by its number of sides. Then classify it as convex or concave and regular orirregular.
6. Lou is roping a boundary for an event at a local carnival. She has 144 feet of rope, and the eventsupervisor has instructed her to rope an area that is three times as long as it is wide. Find the length of each roped side.
7. Make a conjecture about the next letter in the sequence.L M N P Q R T …
8. Construct a truth table for �p � q.
9. Identify the hypothesis and conclusion of the statement. In aplane, if lines � and m are equidistant from line p, then � || m .
10. Cailyn knows that if two angles are vertical, they are congruent.She also knows that if two angles are congruent, then theyhave the same measure. She is given vertical angles 1 and 2,and she concludes that m�1 � m�2. What law of reasoningdoes she use?
11. Determine whether the following statement is always,sometimes, or never true. Points X, Y, and Z determine two lines.
3xx
6x�2x�
P TQ
RS
x�
H J K L
11–8 in. 35–
8 in.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 182 Glencoe Geometry
Unit 1 Review (continued)33
For Questions 12–16, complete the proof.
Given: �1 and �2 are complementary.�3 and �4 are complementary.�1 � �3
Prove: �3 and �2 are complementary.
Statements Reasons1. �1 and �2 are complementary. 1.
�3 and �4 are complementary.�1 � �3
2. �2 � �4 2.3. m�2 � m�4 3.4. m�3 � m�4 � 90 4.5. m�3 � m�2 � 90 5.6. �3 and �2 are complementary 6. Def. of comp. �
For Questions 17 and 18, refer to the figure.17. Name the transversal that forms
�3 and �6. Then identify the specialname for the angle pair.
18. If p || q, m�1 � 5b � 23, and m�11 � 2b � 10, find m�2,m�4, m�10, and m�12.
19. Determine whether QR��� and ST��� are parallel, perpendicular, orneither for Q(�4, �4), R(5, 2), S(4, �5), and T(0, 1).
20. Graph the line that contains C(3, �1) and has a slope of �2.Then write an equation for the line in slope-intercept form.
21. Find x so that k || �.
22. Find the distance between two lines that have equations y � 3x � 1 and y � 3x � 19.
�
k(3x � 11)�
(4x � 1)�
m
q
�p 1 2
39 10
11 12
45 67 8
(Question 16)(Question 15)(Question 14)(Question 13)
(Question 12)
1 42 3
NAME DATE PERIOD
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Standardized Test PracticeStudent Record Sheet (Use with pages 172–173 of the Student Edition.)
33
© Glencoe/McGraw-Hill A1 Glencoe Geometry
An
swer
s
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7
2 5 8
3 6 9 DCBADCBADCBA
DCBADCBADCBA
DCBADCBADCBA
NAME DATE PERIOD
Part 1 Multiple ChoicePart 1 Multiple Choice
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 3 Open-EndedPart 3 Open-Ended
Solve the problem and write your answer in the blank.
For Questions 11, 12, and 13, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.
10 11 12 13
11 (grid in)
12 (grid in)
13 (grid in)
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
Record your answers for Questions 14–15 on the back of this paper.
© Glencoe/McGraw-Hill A2 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Par
alle
l Lin
es a
nd
Tra
nsv
ersa
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-1
3-1
©G
lenc
oe/M
cGra
w-H
ill12
5G
lenc
oe G
eom
etry
Lesson 3-1
Rel
atio
nsh
ips
Bet
wee
n L
ines
an
d P
lan
esW
hen
tw
o li
nes
lie
in
th
e sa
me
plan
e an
d do
not
in
ters
ect,
they
are
par
alle
l.L
ines
th
at d
o n
ot i
nte
rsec
t an
d ar
e n
ot c
opla
nar
are
sk
ew l
ines
.In
th
e fi
gure
, �is
par
alle
l to
m,o
r �
|| m.Y
ou c
an a
lso
wri
te
PQ
� �� ||
RS
� �� .
Sim
ilar
ly,i
f tw
o pl
anes
do
not
in
ters
ect,
they
are
p
aral
lel
pla
nes
.
a.N
ame
all
pla
nes
th
at a
re p
aral
lel
to p
lan
e A
BD
.pl
ane
EF
H
b.
Nam
e al
l se
gmen
ts t
hat
are
par
alle
l to
C �G�
.B �
F�,D�
H�,a
nd
A�E�
c.N
ame
all
segm
ents
th
at a
re s
kew
to
E�H�
.B �
F�,C�
G�,B�
D�,C�
D�,a
nd
A�B�
For
Exe
rcis
es 1
–3,r
efer
to
the
figu
re a
t th
e ri
ght.
1.N
ame
all
plan
es t
hat
in
ters
ect
plan
e O
PT
.
MN
O,M
PS
,NO
T,R
ST
2.N
ame
all
segm
ents
th
at a
re p
aral
lel
to N�
U�.
O�T�,
P�S�
,M�R�
3.N
ame
all
segm
ents
th
at i
nte
rsec
t M�
P�.
M�R�
,M�N�
,M�S�
,P�S�
,P�O�
For
Exe
rcis
es 4
–7,r
efer
to
the
figu
re a
t th
e ri
ght.
4.N
ame
all
segm
ents
par
alle
l to
Q�X�
.
R�A�
,S�G�
,T�O�
,M�H�
,N�E�
5.N
ame
all
plan
es t
hat
in
ters
ect
plan
e M
HE
.
MH
O,N
EX
,HE
X,M
NQ
,SG
O,R
AX
6.N
ame
all
segm
ents
par
alle
l to
Q�R�
.
A�X�
,H�O�
,M�T�
7.N
ame
all
segm
ents
ske
w t
o A�
G�.
S�T�,
T�M�,N�
Q�,Q�
R�,T�
O�,M�
H�,N�
E�,Q�
X�
T
SG
A
X
EN
MH
O
Q
R
M
NO
P
R
UT
S
A
BC
D
E
FG
H
PQ
RSn
m�
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill12
6G
lenc
oe G
eom
etry
An
gle
Rel
atio
nsh
ips
A l
ine
that
in
ters
ects
tw
o or
mor
e ot
her
lin
es i
n a
pla
ne
is c
alle
da
tran
sver
sal.
In t
he
figu
re b
elow
,tis
a t
ran
sver
sal.
Tw
o li
nes
an
d a
tran
sver
sal
form
eig
ht
angl
es.S
ome
pair
s of
th
e an
gles
hav
e sp
ecia
l n
ames
.Th
e fo
llow
ing
char
t li
sts
the
pair
s of
angl
es a
nd
thei
r n
ames
.
An
gle
Pai
rsN
ame
�3,
�4,
�5,
and
�6
inte
rior
angl
es
�3
and
�5;
�4
and
�6
alte
rnat
e in
terio
r an
gles
�3
and
�6;
�4
and
�5
cons
ecut
ive
inte
rior
angl
es
�1,
�2,
�7,
and
�8
exte
rior
angl
es
�1
and
�7;
�2
and
�8
alte
rnat
e ex
terio
r an
gles
�1
and
�5;
�2
and
�6;
�
3 an
d �
7; �
4 an
d �
8co
rres
pond
ing
angl
es
Iden
tify
eac
h p
air
of a
ngl
es a
s a
lter
na
te
inte
rior
,alt
ern
ate
ext
erio
r,co
rres
pon
din
g,or
con
secu
tive
inte
rior
angl
es.
a.�
10 a
nd
�16
b.
�4
and
�12
alte
rnat
e ex
teri
or a
ngl
esco
rres
pon
din
g an
gles
c.�
12 a
nd
�13
d.
�3
and
�9
con
secu
tive
in
teri
or a
ngl
esal
tern
ate
inte
rior
an
gles
Use
th
e fi
gure
in
th
e E
xam
ple
for
Exe
rcis
es 1
–12.
Nam
e th
e tr
ansv
ersa
l th
at f
orm
s ea
ch p
air
of a
ngl
es.
1.�
9 an
d �
13 q
2.�
5 an
d �
14 �
3.�
4 an
d �
6 p
Iden
tify
eac
h p
air
of a
ngl
es a
s a
lter
na
te i
nte
rior
,alt
ern
ate
ext
erio
r,co
rres
pon
din
g,or
con
secu
tive
in
teri
oran
gles
.
4.�
1 an
d �
55.
�6
and
�14
6.�
2 an
d �
8
corr
esp
on
din
gco
rres
po
nd
ing
alt.
exte
rio
r
7.�
3 an
d �
118.
�12
an
d �
39.
�4
and
�6
corr
esp
on
din
gco
nse
cuti
ve in
teri
or
alt.
inte
rio
r
10.�
6 an
d �
1611
.�11
an
d �
1412
.�10
an
d �
16
alt.
inte
rio
rco
nse
cuti
ve in
teri
or
alt.
exte
rio
r
�
pq
n1
29
10 1112
34
1413
65
1516
78
t
nm1
2 34 5
6 78
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Par
alle
l Lin
es a
nd
Tra
nsv
ersa
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-1
3-1
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 3-1)
© Glencoe/McGraw-Hill A3 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Par
alle
l Lin
es a
nd
Tra
nsv
ersa
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-1
3-1
©G
lenc
oe/M
cGra
w-H
ill12
7G
lenc
oe G
eom
etry
Lesson 3-1
For
Exe
rcis
es 1
–4,r
efer
to
the
figu
re a
t th
e ri
ght.
1.N
ame
all
plan
es t
hat
are
par
alle
l to
pla
ne
DE
H.
pla
ne
FB
G
2.N
ame
all
segm
ents
th
at a
re p
aral
lel
to A�
B�.
C�D�
,E�F�,
G�H�
3.N
ame
all
segm
ents
th
at i
nte
rsec
t G�
H�.
C�G�
,D�H�
,E�H�
,F�G�
4.N
ame
all
segm
ents
th
at a
re s
kew
to
C�D�
.A�
E�,B�
F�,E�
H�,F�
G�
Iden
tify
th
e se
ts o
f li
nes
to
wh
ich
eac
h g
iven
lin
e is
a
tran
sver
sal.
5.r
san
d t
,san
d w
,san
d z
,tan
d w
,tan
d z
,wan
d z
6.s
ran
d t
,ran
d w
,ran
d z
,tan
d w
,tan
d z
,wan
d z
7.w
ran
d s
,ran
d t
,ran
d z
,san
d t
,san
d z
,tan
d z
Iden
tify
eac
h p
air
of a
ngl
es a
s a
lter
na
te i
nte
rior
,alt
ern
ate
ex
teri
or,c
orre
spon
din
g,or
con
secu
tive
in
teri
oran
gles
.
8.�
2 an
d �
89.
�3
and
�6
alte
rnat
e ex
teri
or
con
secu
tive
inte
rio
r
10.�
1 an
d �
911
.�3
and
�9
corr
esp
on
din
gal
tern
ate
inte
rio
r
12.�
6 an
d �
1213
.�7
and
�11
alte
rnat
e ex
teri
or
corr
esp
on
din
g
Nam
e th
e tr
ansv
ersa
l th
at f
orm
s ea
ch p
air
of a
ngl
es.T
hen
id
enti
fy t
he
spec
ial
nam
e fo
r th
e an
gle
pai
r.
14.�
4 an
d �
1015
.�2
and
�12
g;al
tern
ate
inte
rio
rg;
alte
rnat
e ex
teri
or
16.�
7 an
d �
317
.�13
an
d �
10
j;co
rres
po
nd
ing
k;
con
secu
tive
inte
rio
r
18.�
8 an
d �
1419
.�6
and
�14
h;al
tern
ate
inte
rio
rh;
corr
esp
on
din
g
h
g
j
k
12
91011
12
1315
14 163
4
65
78
a
b c d
12
910
1112
34
65
78
r st
w
z
A
EF
B
D
HG
C
©G
lenc
oe/M
cGra
w-H
ill12
8G
lenc
oe G
eom
etry
For
Exe
rcis
es 1
–4,r
efer
to
the
figu
re a
t th
e ri
ght.
1.N
ame
all
plan
es t
hat
in
ters
ect
plan
e S
TX
.T
UY
,RS
W,S
TU
,VW
X,Q
UV,
QV
W2.
Nam
e al
l se
gmen
ts t
hat
in
ters
ect
Q�U�
.Q�
R�,Q�
V�,T�
U�,U�
Z�
3.N
ame
all
segm
ents
th
at a
re p
aral
lel
to X�
Y�.
S�T�
4.N
ame
all
segm
ents
th
at a
re s
kew
to
V�W�
.Q�
U�,R�
S�,S�
T�,S�
X�,T�
U�,T�
Y�,U�
Z�
Iden
tify
th
e se
ts o
f li
nes
to
wh
ich
eac
h g
iven
lin
e is
a
tran
sver
sal.
5.e f
and
g,f
and
h,f
and
i,g
and
h,g
and
i,h
and
i
6.h e
and
f,e
and
g,e
and
i,f
and
g,f
and
i,g
and
i
Iden
tify
eac
h p
air
of a
ngl
es a
s a
lter
na
te i
nte
rior
,alt
ern
ate
ex
teri
or,c
orre
spon
din
g,or
con
secu
tive
in
teri
oran
gles
.
7.�
9 an
d �
138.
�6
and
�16
corr
esp
on
din
gal
tern
ate
exte
rio
r
9.�
3 an
d �
1010
.�8
and
�14
con
secu
tive
inte
rio
ral
tern
ate
inte
rio
r
Nam
e th
e tr
ansv
ersa
l th
at f
orm
s ea
ch p
air
of a
ngl
es.T
hen
id
enti
fy t
he
spec
ial
nam
e fo
r th
e an
gle
pai
r.
11.�
2 an
d �
1212
.�6
and
�18
a;al
tern
ate
inte
rio
rd;
corr
esp
on
din
g
13.�
13 a
nd
�19
14.�
11 a
nd
�7
b;al
tern
ate
exte
rio
rc;
con
secu
tive
inte
rio
r
FUR
NIT
UR
EF
or E
xerc
ises
15–
16,r
efer
to
the
dra
win
g of
th
e en
d t
able
.
15.F
ind
an e
xam
ple
of p
aral
lel
plan
es.
Sam
ple
an
swer
:th
e to
p
of
the
tab
le a
nd
th
e b
ott
om
sh
elf
16.F
ind
an e
xam
ple
of p
aral
lel
lin
es.
Sam
ple
an
swer
:th
e ta
ble
leg
scd
b
a1
910
1112
1314
1516
1719
18 20
56
78
23
4
m
�
pn
1 910 11
1213
1415
16
56
78
2 34
g
he
i
f
QR
ST
U
VW
X
Y
Z
Pra
ctic
e (
Ave
rag
e)
Par
alle
l Lin
es a
nd
Tra
nsv
ersa
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-1
3-1
Answers (Lesson 3-1)
© Glencoe/McGraw-Hill A4 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csP
aral
lel L
ines
an
d T
ran
sver
sals
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-1
3-1
©G
lenc
oe/M
cGra
w-H
ill12
9G
lenc
oe G
eom
etry
Lesson 3-1
Pre-
Act
ivit
yH
ow a
re p
aral
lel
lin
es a
nd
pla
nes
use
d i
n a
rch
itec
ture
?
Rea
d th
e in
trod
uct
ion
to
Les
son
3-1
at
the
top
of p
age
126
in y
our
text
book
.
•G
ive
an e
xam
ple
of p
aral
lel
lin
es t
hat
can
be
fou
nd
in y
our
clas
sroo
m.
Sam
ple
an
swer
s:ed
ges
of
flo
or
alo
ng
op
po
site
wal
ls;
vert
ical
ed
ges
of
a d
oo
r•
Giv
e an
exa
mpl
e of
par
alle
l pl
anes
th
at c
an b
e fo
un
d in
you
r cl
assr
oom
.S
amp
le a
nsw
ers:
ceili
ng
an
d f
loo
r;o
pp
osi
te w
alls
Rea
din
g t
he
Less
on
1.W
rite
a g
eom
etri
cal
term
th
at m
atch
es e
ach
def
init
ion
.
a.tw
o pl
anes
th
at d
o n
ot i
nte
rsec
tp
aral
lel p
lan
esb
.li
nes
th
at a
re n
ot c
opla
nar
an
d do
not
in
ters
ect
skew
lin
esc.
two
copl
anar
lin
es t
hat
do
not
in
ters
ect
par
alle
l lin
esd
.a
lin
e th
at i
nte
rsec
ts t
wo
or m
ore
lin
es i
n a
pla
ne
at d
iffe
ren
t po
ints
tran
sver
sal
e.a
pair
of
angl
es d
eter
min
ed b
y tw
o li
nes
an
d a
tran
sver
sal
con
sist
ing
of a
n i
nte
rior
angl
e an
d an
ext
erio
r an
gle
that
hav
e di
ffer
ent
vert
ices
an
d th
at l
ie o
n t
he
sam
e si
deof
th
e tr
ansv
ersa
lco
rres
po
nd
ing
an
gle
s
2.R
efer
to
the
figu
re a
t th
e ri
ght.
Giv
e th
e sp
ecia
l n
ame
for
each
an
gle
pair
.
a.�
3 an
d �
5co
rres
po
nd
ing
an
gle
sb
.�
6 an
d �
12al
tern
ate
exte
rio
r an
gle
sc.
�4
and
�8
alte
rnat
e in
teri
or
ang
les
d.
�2
and
�3
con
secu
tive
inte
rio
r an
gle
se.
�8
and
�12
corr
esp
on
din
g a
ng
les
f.�
5 an
d �
9al
tern
ate
inte
rio
r an
gle
sg.
�4
and
�10
vert
ical
an
gle
sh
.�
6 an
d �
7lin
ear
pai
r
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
new
mat
hem
atic
al t
erm
s is
to
rela
te t
hem
to
wor
ds t
hat
you
use
in
eve
ryda
y li
fe.M
any
wor
ds s
tart
wit
h t
he
pref
ix t
ran
s-,w
hic
h i
s a
Lat
in r
oot
mea
nin
g ac
ross
.Lis
t fo
ur
En
glis
h w
ords
th
at s
tart
wit
h t
ran
s-.H
ow c
an t
he
mea
nin
g of
this
pre
fix
hel
p yo
u r
emem
ber
the
mea
nin
g of
tra
nsv
ersa
l?S
amp
le a
nsw
er:T
ran
slat
e,tr
ansf
er,t
ran
spo
rt,t
ran
sco
nti
nen
tal;
atr
ansv
ersa
l is
a lin
e th
at g
oes
acr
oss
tw
o o
r m
ore
oth
er li
nes
.
� m np
12
910
1112
3 4
65
78
©G
lenc
oe/M
cGra
w-H
ill13
0G
lenc
oe G
eom
etry
Per
spec
tive
Dra
win
gs
To
draw
th
ree-
dim
ensi
onal
ob
ject
s,ar
tist
s m
ake
per
spec
tive
dra
win
gssu
ch a
s th
e on
es s
how
n.
To
indi
cate
dep
th i
n a
pers
pect
ive
draw
ing,
som
epa
rall
el l
ines
are
dra
wn
as
con
verg
ing
lin
es.T
he
dott
ed l
ines
in
th
e fi
gure
sbe
low
eac
h e
xten
d to
ava
nis
hin
g p
oin
t,or
spo
tw
her
e pa
rall
el l
ines
ap
pear
to
mee
t.
Dra
w l
ines
to
loca
te t
he
van
ish
ing
poi
nt
in e
ach
dra
win
g of
a b
ox.
1.2.
3.
4.T
he
fron
ts o
f tw
o cu
bes
are
show
n b
elow
.Usi
ng
poin
t P
as t
he
van
ish
ing
poin
t fo
r bo
th c
ube
s,co
mpl
ete
the
pers
pect
ive
draw
ings
of
the
cube
s.
5.F
ind
an e
xam
ple
of a
per
spec
tive
dra
win
g in
a n
ewsp
aper
or
mag
azin
e.T
race
th
e dr
awin
g an
d lo
cate
a v
anis
hin
g po
int.
See
stu
den
ts’w
ork
.
P
Vani
shin
g po
ints
Railr
oad
track
sCu
beCa
bine
t
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-1
3-1
Answers (Lesson 3-1)
© Glencoe/McGraw-Hill A5 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
An
gle
s an
d P
aral
lel L
ines
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-2
3-2
©G
lenc
oe/M
cGra
w-H
ill13
1G
lenc
oe G
eom
etry
Lesson 3-2
Para
llel L
ines
an
d A
ng
le P
airs
Wh
en t
wo
para
llel
lin
es a
re c
ut
by a
tra
nsv
ersa
l,th
e fo
llow
ing
pair
s of
an
gles
are
con
gru
ent.
•co
rres
pon
din
g an
gles
•al
tern
ate
inte
rior
an
gles
•al
tern
ate
exte
rior
an
gles
Als
o,co
nse
cuti
ve i
nte
rior
an
gles
are
su
pple
men
tary
.
In t
he
figu
re,m
�2
�75
.Fin
d t
he
mea
sure
s of
th
e re
mai
nin
g an
gles
.m
�1
�10
5�
1 an
d �
2 fo
rm a
lin
ear
pair
.m
�3
�10
5�
3 an
d �
2 fo
rm a
lin
ear
pair
.m
�4
�75
�4
and
�2
are
vert
ical
an
gles
.m
�5
�10
5�
5 an
d �
3 ar
e al
tern
ate
inte
rior
an
gles
.m
�6
�75
�6
and
�2
are
corr
espo
ndi
ng
angl
es.
m�
7 �
105
�7
and
�3
are
corr
espo
ndi
ng
angl
es.
m�
8 �
75�
8 an
d �
6 ar
e ve
rtic
al a
ngl
es.
In t
he
figu
re,m
�3
�10
2.F
ind
th
e m
easu
re o
f ea
ch a
ngl
e.
1.�
510
22.
�6
78
3.�
11 1
024.
�7
102
5.�
1510
26.
�14
78
In t
he
figu
re,m
�9
�80
an
d m
�5
�68
.Fin
d t
he
mea
sure
of
eac
h a
ngl
e.
7.�
1210
08.
�1
80
9.�
410
010
.�3
80
11.�
768
12.�
1611
2
wv
p q
12 3
4
65
78
910 11
12
1413
1516
pq
m n
12 3
4
65
78
910 11
12
1413
1516
p
m n
12 3
4 65
78
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill13
2G
lenc
oe G
eom
etry
Alg
ebra
an
d A
ng
le M
easu
res
Alg
ebra
can
be
use
d to
fi
nd
un
know
n v
alu
es i
n a
ngl
es f
orm
ed b
y a
tran
sver
sal
and
para
llel
lin
es.
If m
�1
�3x
�15
,m�
2 �
4x�
5,m
�3
�5y
,an
d m
�4
�6z
�3,
fin
d x
and
y.
p||q
,so
m�
1 �
m�
2 be
cau
se t
hey
are
co
rres
pon
din
g an
gles
.
3x�
15 �
4x�
53x
�15
�3x
�4x
�5
�3x
15 �
x�
515
�5
�x
�5
�5
20 �
x
pq
r s
12
34
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
An
gle
s an
d P
aral
lel L
ines
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-2
3-2
Exam
ple
Exam
ple
r||s
,so
m�
2 �
m�
3 be
cau
se t
hey
are
co
rres
pon
din
g an
gles
.
m�
2 �
m�
375
�5y
�7 55 ��
�5 5y �
15 �
y
Exer
cises
Exer
cises
Fin
d x
and
yin
eac
h f
igu
re.
1.2.
x�
15;
y�
19
x�
6;y
�24
3.4.
x�
11;
y�
10x
�10
;y
�25
Fin
d x
,y,a
nd
zin
eac
h f
igu
re.
5.6.
x�
74;
y�
37;
z�
25
x�
30;
y�
15;
z�
150
2x�
2y�
90�
x�
z�
2y�
106�
x�( 4
z �
6) �
( 5x
� 2
0)�
3 x�
2 y� 4y
�
( 11x
� 4
) �
( 13y
� 5
) �( 5y
� 5
) �
5x�
( 15x
� 3
0)�
( 3y
� 1
8)�
10x�
90�
( 5x
� 5
) �( 6
y �
4) �
( 4x
� 1
0)�
Answers (Lesson 3-2)
© Glencoe/McGraw-Hill A6 Glencoe Geometry
Skil
ls P
ract
ice
An
gle
s an
d P
aral
lel L
ines
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-2
3-2
©G
lenc
oe/M
cGra
w-H
ill13
3G
lenc
oe G
eom
etry
Lesson 3-2
In t
he
figu
re,m
�2
�70
.Fin
d t
he
mea
sure
of
each
an
gle.
1.�
370
2.�
511
0
3.�
811
04.
�1
110
5.�
411
06.
�6
70
In t
he
figu
re,m
�7
�10
0.F
ind
th
e m
easu
re o
f ea
ch a
ngl
e.
7.�
910
08.
�6
80
9.�
880
10.�
280
11.�
510
012
.�11
100
In t
he
figu
re,m
�3
�75
an
d m
�10
�11
5.F
ind
th
e m
easu
re
of e
ach
an
gle.
13.�
210
514
.�5
105
15.�
710
516
.�15
115
17.�
1465
18.�
965
Fin
d x
and
yin
eac
h f
igu
re.
19.
x�
28,y
�47
20.
x�
10,y
�15
Fin
d m
�1
in e
ach
fig
ure
.
21.
8022
.72
140�
32�
160
� 1 20�
7x�
( 8x
� 1
0)�
( 6y
� 2
0)�
5x�
40�
( 3y
� 1
) �
x
w
z
y
12
43
65
8
910
1112
1314
1516
7sm
ut
12 43
65 8
910 11
127
q
r s
12 4
3 65
87
©G
lenc
oe/M
cGra
w-H
ill13
4G
lenc
oe G
eom
etry
In t
he
figu
re,m
�2
�92
an
d m
�12
�74
.Fin
d t
he
mea
sure
of
eac
h a
ngl
e.
1.�
1092
2.�
892
3.�
988
4.�
510
6
5.�
1110
66.
�13
106
Fin
d x
and
yin
eac
h f
igu
re.
7.8.
x�
14,y
�37
x�
28,y
�23
Fin
d m
�1
in e
ach
fig
ure
.
9.10
.
130
98
11.P
RO
OF
Wri
te a
par
agra
ph p
roof
of T
heo
rem
3.3
.G
iven
: �|| m
,m|| n
Pro
ve:�
1 �
�12
Sam
ple
pro
of:
It is
giv
en t
hat
�|| m
,so
�1
��
8 by
th
e A
lter
nat
eE
xter
ior
An
gle
s T
heo
rem
.Sin
ce it
is g
iven
th
at m
|| n,
�8
��
12 b
y th
e C
orr
esp
on
din
g A
ng
les
Po
stu
late
.T
her
efo
re,�
1 �
�12
,sin
ce c
on
gru
ence
of
ang
les
is t
ran
siti
ve.
12.F
ENC
ING
A d
iago
nal
bra
ce s
tren
gth
ens
the
wir
e fe
nce
an
d pr
even
ts
it f
rom
sag
gin
g.T
he
brac
e m
akes
a 5
0°an
gle
wit
h t
he
wir
e as
sh
own
.F
ind
y.13
050
�
y�
m n
12
34
65 7
8
�
k
910
1112
144�
62� 1
100�
50�
1
3y�
( 2x
� 1
3)�
( 5y
� 4
) �3x
�( 9x
� 1
2)�
( 4y
� 1
0)�
n
m
rs
12
43 6
5
8
910
1112
1314
1516
7
Pra
ctic
e (
Ave
rag
e)
An
gle
s an
d P
aral
lel L
ines
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-2
3-2
Answers (Lesson 3-2)
© Glencoe/McGraw-Hill A7 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csA
ng
les
and
Par
alle
l Lin
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-2
3-2
©G
lenc
oe/M
cGra
w-H
ill13
5G
lenc
oe G
eom
etry
Lesson 3-2
Pre-
Act
ivit
yH
ow c
an a
ngl
es a
nd
lin
es b
e u
sed
in
art
?
Rea
d th
e in
trod
uct
ion
to
Les
son
3-2
at
the
top
of p
age
133
in y
our
text
book
.•
You
r te
xtbo
ok s
how
s a
pain
tin
g th
at c
onta
ins
two
para
llel
lin
es a
nd
atr
ansv
ersa
l.W
hat
is t
he n
ame
for
�1
and
�2?
corr
esp
on
din
g a
ng
les
•W
hat
is
the
rela
tion
ship
bet
wee
n t
hes
e tw
o an
gles
?T
hey
are
co
ng
ruen
t.
Rea
din
g t
he
Less
on
1.C
hoo
se t
he
corr
ect
wor
d to
com
plet
e ea
ch s
ente
nce
.
a.If
tw
o pa
rall
el l
ines
are
cu
t by
a t
ran
sver
sal,
then
alt
ern
ate
exte
rior
an
gles
are
(con
gru
ent/
com
plem
enta
ry/s
upp
lem
enta
ry).
b.
If t
wo
para
llel
lin
es a
re c
ut
by a
tra
nsv
ersa
l,th
en c
orre
spon
din
g an
gles
are
(con
gru
ent/
com
plem
enta
ry/s
upp
lem
enta
ry).
c.If
par
alle
l li
nes
are
cu
t by
a t
ran
sver
sal,
then
con
secu
tive
in
teri
or a
ngl
es a
re
(con
gru
ent/
com
plem
enta
ry/s
upp
lem
enta
ry).
d.
In a
pla
ne,
if a
lin
e is
per
pen
dicu
lar
to o
ne
of t
wo
para
llel
lin
es,t
hen
it
is
(par
alle
l/per
pen
dicu
lar/
skew
) to
th
e ot
her
.
Use
th
e fi
gure
for
Exe
rcis
es 2
an
d 3
.
2.a.
Nam
e fo
ur
pair
s of
ver
tica
l an
gles
.�
1 an
d �
3,�
2 an
d �
4,�
5 an
d �
7,�
6 an
d �
8b
.N
ame
all
angl
es t
hat
for
m a
lin
ear
pair
wit
h �
7.�
6,�
8c.
Nam
e al
l an
gles
th
at a
re c
ongr
uen
t to
�1.
�3,
�6,
�8
d.
Nam
e al
l an
gles
th
at a
re c
ongr
uen
t to
�4.
�2,
�5,
�7
e.N
ame
all
angl
es t
hat
are
su
pple
men
tary
to
�3.
�2,
�4,
�5,
�7
f.N
ame
all
angl
es t
hat
are
su
pple
men
tary
to
�2.
�1,
�3,
�6,
�8
3.W
hic
h c
oncl
usi
on(s
) co
uld
you
mak
e ab
out
lin
es u
and
vif
m�
4 �
m�
1?B
,DA
.t
|| uB
.t
⊥u
C.
v⊥
uD
.v
⊥t
E.
v|| t
Hel
pin
g Y
ou
Rem
emb
er4.
How
can
you
use
an
eve
ryda
y m
ean
ing
of t
he
adje
ctiv
e al
tern
ate
to h
elp
you
rem
embe
rth
e ty
pes
of a
ngl
e pa
irs
for
two
lin
es a
nd
a tr
ansv
ersa
l?
Sam
ple
an
swer
:O
ne
mea
nin
g o
f al
tern
ate
is “
ob
tain
ed b
y sw
itch
ing
bac
kan
d f
ort
h f
rom
on
e th
ing
to
an
oth
er.”
Th
e an
gle
pai
rs in
th
is le
sso
n a
llu
se a
ng
les
wit
h d
iffe
ren
t ve
rtic
es,a
nd
th
ose
wh
ose
nam
es c
on
tain
th
ead
ject
ive
alte
rnat
eca
n b
e lo
cate
d in
a f
igu
re b
y sw
itch
ing
fro
m o
ne
sid
eo
f th
e tr
ansv
ersa
l to
th
e o
ther
.Th
e p
airs
wh
ose
nam
es d
o n
ot
incl
ud
e th
ew
ord
alt
ern
ate
are
fou
nd
on
th
e sa
me
sid
e o
f th
e tr
ansv
ersa
l.
t
u
v
1 234
6578
per
pen
dic
ula
r
sup
ple
men
tary
con
gru
ent
con
gru
ent
©G
lenc
oe/M
cGra
w-H
ill13
6G
lenc
oe G
eom
etry
Mo
re O
pti
cal I
llusi
on
sIn
dra
win
gs,d
iago
nal
lin
es m
ay c
reat
e th
e il
lusi
on o
f de
pth
.F
or e
xam
ple,
the
figu
re a
t th
e ri
ght
can
be
thou
ght
of a
s pi
ctu
rin
g a
flat
fig
ure
or
a cu
be.T
he
opti
cal
illu
sion
s on
th
is
page
in
volv
e de
pth
per
cept
ion
.
An
swer
eac
h q
ues
tion
.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-2
3-2
1.H
ow m
any
cube
s do
you
see
in
th
edr
awin
g?5
or
6
3.D
oes
the
draw
ing
show
a v
iew
fro
m
the
top
or t
he
bott
om o
f th
e st
airs
?
An
swer
s m
ay v
ary.
2.C
an t
his
fig
ure
sh
ow a
n a
ctu
al
obje
ct?
no
4.W
hic
h l
ine
segm
ent
is l
onge
r,A�
B�or
C �D�
? M
easu
re t
o ch
eck
you
r an
swer
.
A�B�
A
BC
D
5.W
hic
h p
erso
n i
n t
he
draw
ing
at t
he
righ
t ap
pear
s to
be
tall
est?
Mea
sure
to
chec
k yo
ur
answ
er.
Th
e p
erso
n a
t th
e ri
gh
t ap
pea
rs t
alle
st,
but
all a
re t
he
sam
e si
ze.
6.D
raw
tw
o m
ore
obje
cts
the
sam
e si
ze o
n t
he
figu
re
at t
he
righ
t.D
oes
one
appe
ar l
arge
r th
an t
he
oth
er?
An
swer
s w
ill v
ary.
Answers (Lesson 3-2)
© Glencoe/McGraw-Hill A8 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Slo
pes
of
Lin
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-3
3-3
©G
lenc
oe/M
cGra
w-H
ill13
7G
lenc
oe G
eom
etry
Lesson 3-3
Slo
pe
of
a Li
ne
Th
e sl
ope
mof
a l
ine
con
tain
ing
two
poin
ts w
ith
coo
rdin
ates
(x 1
,y1)
and
(x2,
y 2)
is g
iven
by
the
form
ula
m�
�y x2 2
� �
y x1 1�
,wh
ere
x 1�
x 2.
Fin
d t
he
slop
e of
eac
h l
ine.
For
lin
e p,
let
(x1,
y 1)
be (
1,2)
an
d (x
2,y 2
) be
(�
2,�
2).
m�
�y x2 2
� �
y x1 1�
��� �
2 2� �
2 1�
or �4 3�
For
lin
e q,
let
(x1,
y 1)
be (
2,0)
an
d (x
2,y 2
) be
(�
3,2)
.
m�
�y x2 2
� �
y x1 1�
�� �2 3� �
0 2�
or �
�2 5�
Det
erm
ine
the
slop
e of
th
e li
ne
that
con
tain
s th
e gi
ven
poi
nts
.
1.J
(0,0
),K
(�2,
8)�
42.
R(�
2,�
3),S
(3,�
5)�
�2 5�
3.L
(1,�
2),N
(�6,
3)�
�5 7�4.
P(�
1,2)
,Q(�
9,6)
��1 2�
5.T
(1,�
2),U
(6,�
2)0
6.V
(�2,
10),
W(�
4,�
3)�1 23 �
Fin
d t
he
slop
e of
eac
h l
ine.
7.A
B��
�3
8.C
D��
��
2
9.E
M��
�u
nd
efin
ed10
.AE
���
0
11.E
H��
��2 5�
12.B
M��
��
�1 2�
x
y
O
C( –
2, 2
)
A( –
2, –
2) H
( –1,
–4)
B( 0
, 4)
M( 4
, 2)
E( 4
, –2)
D( 0
, –2)
x
y
O
( 1, 2
)
( –2,
–2)( –
3, 2
)
( 2, 0
)
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill13
8G
lenc
oe G
eom
etry
Para
llel a
nd
Per
pen
dic
ula
r Li
nes
If y
ou e
xam
ine
the
slop
es o
f pa
irs
of p
aral
lel
lin
esan
d th
e sl
opes
of
pair
s of
per
pen
dicu
lar
lin
es,w
her
e n
eith
er l
ine
in e
ach
pai
r is
ver
tica
l,yo
uw
ill
disc
over
th
e fo
llow
ing
prop
erti
es.
Tw
o li
nes
hav
e th
e sa
me
slop
e if
an
d on
ly i
f th
ey a
re p
aral
lel.
Tw
o li
nes
are
per
pen
dicu
lar
if a
nd
only
if
the
prod
uct
of
thei
r sl
opes
is
�1.
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Slo
pes
of
Lin
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-3
3-3
Fin
d t
he
slop
e of
a l
ine
par
alle
l to
th
e li
ne
con
tain
ing
A(�
3,4)
and
B(2
,5).
Fin
d th
e sl
ope
of A
B� �
� .U
se (
�3,
4) f
or (
x 1,y
1)an
d u
se (
2,5)
for
(x 2
,y2)
.
m�
�y x2 2
� �
y x1 1�
�� 2
5 �
� (�4 3)
�or
�1 5�
Th
e sl
ope
of a
ny
lin
e pa
rall
el t
o A
B��
�m
ust
be
�1 5� .
Fin
d t
he
slop
e of
a l
ine
per
pen
dic
ula
r to
PQ
� ��
for
P(�
2,�
4) a
nd
Q(4
,3).
Fin
d th
e sl
ope
of P
Q� �
� .U
se (
�2,
�4)
for
(x
1,y 1
) an
d u
se (
4,3)
for
(x 2
,y2)
.
m�
�y x2 2
� �
y x1 1�
��3 4
� �
( (� �
4 2) )�
or �7 6�
Sin
ce �7 6�
���
�6 7� ���
1,th
e sl
ope
of a
ny
lin
e
perp
endi
cula
r to
PQ
��� m
ust
be
��6 7� .
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Det
erm
ine
wh
eth
er M
N��
�an
d R
S��
� are
pa
rall
el,p
erp
end
icu
lar,
or n
eith
er.
1.M
(0,3
),N
(2,4
),R
(2,1
),S
(8,4
)2.
M(�
1,3)
,N(0
,5),
R(2
,1),
S(6
,�1)
par
alle
lp
erp
end
icu
lar
3.M
(�1,
3),N
(4,4
),R
(3,1
),S
(�2,
2)4.
M(0
,�3)
,N(�
2,�
7),R
(2,1
),S
(0,�
3)
nei
ther
par
alle
l
5.M
(�2,
2),N
(1,�
3),R
(�2,
1),S
(3,4
)6.
M(0
,0),
N(2
,4),
R(2
,1),
S(8
,4)
per
pen
dic
ula
rn
eith
er
Fin
d t
he
slop
e of
MN
���
and
th
e sl
ope
of a
ny
lin
e p
erp
end
icu
lar
to M
N��
� .
7.M
(2,�
4),N
(�2,
�1)
8.M
(1,3
),N
(�1,
5)
��3 4� ;
�4 3��
1;1
9.M
(4,�
2),N
(5,3
)10
.M(2
,�3)
,N(�
4,1)
5;�
�1 5��
�2 3� ;�3 2�
Answers (Lesson 3-3)
© Glencoe/McGraw-Hill A9 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Slo
pes
of
Lin
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-3
3-3
©G
lenc
oe/M
cGra
w-H
ill13
9G
lenc
oe G
eom
etry
Lesson 3-3
Det
erm
ine
the
slop
e of
th
e li
ne
that
con
tain
s th
e gi
ven
poi
nts
.
1.S
(�1,
2),W
(0,4
)2
2.G
(�2,
5),H
(1,�
7)�
4
3.C
(0,1
),D
(3,3
)�2 3�
4.J
(�5,
�2)
,K(5
,�4)
��1 5�
Fin
d t
he
slop
e of
eac
h l
ine.
5.N
P��
�6.
TW
���
�3 4��
2
7.a
lin
e pa
rall
el t
o T
W��
�8.
a li
ne
perp
endi
cula
r to
NP
���
�2
��4 3�
Det
erm
ine
wh
eth
er A
B��
�an
d M
N��
�ar
e p
ara
llel
,per
pen
dic
ula
r,or
nei
ther
.
9.A
(0,3
),B
(5,�
7),M
(�6,
7),N
(�2,
�1)
10.A
(�1,
4),B
(2,�
5),M
(�3,
2),N
(3,0
)p
aral
lel
nei
ther
11.A
(�2,
�7)
,B(4
,2),
M(�
2,0)
,N(2
,6)
12.A
(�4,
�8)
,B(4
,�6)
,M(�
3,5)
,N(�
1,�
3)p
aral
lel
per
pen
dic
ula
r
Gra
ph
th
e li
ne
that
sat
isfi
es e
ach
con
dit
ion
.
13.s
lope
�3,
con
tain
s A
(0,1
)14
.slo
pe �
��3 2� ,
con
tain
s R
(�4,
5)
15.c
onta
ins
Y(3
,0),
para
llel
to
DJ
���
16.c
onta
ins
T(0
,�2)
,per
pen
dicu
lar
to C
X��
�
wit
h D
(�3,
1) a
nd
J(3
,3)
wit
h C
(0,3
) an
d X
(2,�
1)
T( 0
, –2)
C( 0
, 3)
X( 2
, –1)x
y OD
( –3,
1)
J(3,
3)
Y( 3
, 0)
x
y O
R( –
4, 5
)
x
y O
A( 0
, 1)
x
y O
x
y ON
TW
P
©G
lenc
oe/M
cGra
w-H
ill14
0G
lenc
oe G
eom
etry
Det
erm
ine
the
slop
e of
th
e li
ne
that
con
tain
s th
e gi
ven
poi
nts
.
1.B
(�4,
4),R
(0,2
)�
�1 2�2.
I(�
2,�
9),P
(2,4
)�1 43 �
Fin
d t
he
slop
e of
eac
h l
ine.
3.L
M��
�4.
GR
���
�2 3��
�2 5�
5.a
lin
e pa
rall
el t
o G
R��
�6.
a li
ne
perp
endi
cula
r to
PS
���
��2 5�
��1 2�
Det
erm
ine
wh
eth
er K
M��
�an
d S
T��
�ar
e p
ara
llel
,per
pen
dic
ula
r,or
nei
ther
.
7.K
(�1,
�8)
,M(1
,6),
S(�
2,�
6),T
(2,1
0)8.
K(�
5,�
2),M
(5,4
),S
(�3,
6),T
(3,�
4)n
eith
erp
erp
end
icu
lar
9.K
(�4,
10),
M(2
,�8)
,S(1
,2),
T(4
,�7)
10.K
(�3,
�7)
,M(3
,�3)
,S(0
,4),
T(6
,�5)
par
alle
lp
erp
end
icu
lar
Gra
ph
th
e li
ne
that
sat
isfi
es e
ach
con
dit
ion
.
11.s
lope
��
�1 2� ,co
nta
ins
U(2
,�2)
12.s
lope
��4 3� ,
con
tain
s P
(�3,
�3)
13.c
onta
ins
B(�
4,2)
,par
alle
l to
FG
���
14.c
onta
ins
Z(�
3,0)
,per
pen
dicu
lar
to E
K��
�
wit
h F
(0,�
3) a
nd
G(4
,�2)
wit
h E
(�2,
4) a
nd
K(2
,�2)
15. P
RO
FITS
Aft
er T
ake
Tw
o be
gan
ren
tin
g D
VD
s at
th
eir
vide
o st
ore,
busi
nes
s so
ared
.B
etw
een
200
0 an
d 20
03,p
rofi
ts i
ncr
ease
d at
an
ave
rage
rat
e of
$12
,000
per
yea
r.T
otal
prof
its
in 2
003
wer
e $4
6,00
0.If
pro
fits
con
tin
ue
to i
ncr
ease
at
the
sam
e ra
te,w
hat
wil
lth
e to
tal
prof
it b
e in
200
9?$1
18,0
00
K( 2
, –2)
E( –
2, 4
)
Z( –
3, 0
)
x
y
O
F( 0
, –3)
B( –
4, 2
)
G( 4
, –2)
x
y
O
P( –
3, –
3)
x
y OU
( 2, –
2)x
y O
x
y O
R
G
S
P
M
L
Pra
ctic
e (
Ave
rag
e)
Slo
pes
of
Lin
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-3
3-3
Answers (Lesson 3-3)
© Glencoe/McGraw-Hill A10 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csS
lop
es o
f L
ines
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-3
3-3
©G
lenc
oe/M
cGra
w-H
ill14
1G
lenc
oe G
eom
etry
Lesson 3-3
Pre-
Act
ivit
yH
ow i
s sl
ope
use
d i
n t
ran
spor
tati
on?
Rea
d th
e in
trod
uct
ion
to
Les
son
3-3
at
the
top
of p
age
139
in y
our
text
book
.•
If y
ou a
re d
rivi
ng
uph
ill
on a
roa
d w
ith
a 4
% g
rade
,how
man
y fe
et w
ill
the
road
ris
e fo
r ev
ery
1000
hor
izon
tal
feet
tra
vele
d? 4
0 ft
•If
you
are
dri
vin
g do
wn
hil
l on
a r
oad
wit
h a
7%
gra
de,h
ow m
any
met
ers
wil
l th
e ro
ad f
all
for
ever
y 50
0 m
eter
s tr
avel
ed?
35 m
Rea
din
g t
he
Less
on
1.W
hic
h e
xpre
ssio
ns
can
be
use
d to
rep
rese
nt
the
slop
e of
th
e li
ne
con
tain
ing
poin
ts (
x 1,y
1)an
d (x
2,y 2
)? A
ssu
me
that
no
den
omin
ator
is
zero
.A
,C,F
A.
�� �
y x�B
.�h vor ei rz to icn at la rl ir su en
�C
.�y x2 2
� �
y x1 1�
D.�c ch h
a an ng ge e
i in nx y
�
E.�y x2 1
� �
y x1 2�
F.�y x1 1
� �
y x2 2�
G.
�x y2 2
� �
x y1 1�
H.�y y2 1
� �
x x2 1�
2.M
atch
th
e de
scri
ptio
n o
f a
lin
e fr
om t
he
firs
t co
lum
n w
ith
th
e de
scri
ptio
n o
f it
s sl
ope
from
th
e se
con
d co
lum
n.
Typ
e of
Lin
eS
lop
e
a.a
hor
izon
tal
lin
eii
i.a
neg
ativ
e n
um
ber
b.
a li
ne
that
ris
es f
rom
lef
t to
rig
ht
ivii
.0
c.a
vert
ical
lin
eiii
iii.
un
defi
ned
d.
a li
ne
that
fal
ls f
rom
lef
t to
rig
ht
iiv
.a
posi
tive
nu
mbe
r
3.F
ind
the
slop
e of
eac
h l
ine.
a.a
lin
e pa
rall
el t
o a
lin
e w
ith
slo
pe �3 4�
�3 4�
b.
a li
ne
perp
endi
cula
r to
th
e x-
axis
un
def
ined
slo
pe
c.a
lin
e pe
rpen
dicu
lar
to a
lin
e w
ith
slo
pe 5
��1 5�
d.
a li
ne
para
llel
to
the
x-ax
is0
e.y-
axis
un
def
ined
slo
pe
Hel
pin
g Y
ou
Rem
emb
er
4.A
goo
d w
ay t
o re
mem
ber
som
eth
ing
is t
o ex
plai
n i
t to
som
eon
e el
se.S
upp
ose
you
r fr
ien
dth
inks
th
at p
erpe
ndi
cula
r li
nes
(if
nei
ther
lin
e is
ver
tica
l) h
ave
slop
es t
hat
are
reci
proc
als
of e
ach
oth
er.H
ow c
ould
you
exp
lain
to
you
r fr
ien
d th
at t
his
is
inco
rrec
t an
dgi
ve h
er a
goo
d w
ay t
o re
mem
ber
the
corr
ect
rela
tion
ship
?
Sam
ple
an
swer
:In
ord
er f
or
two
lin
es (
nei
ther
on
e ve
rtic
al)
to m
eet
atri
gh
t an
gle
s,o
ne
mu
st g
o u
pw
ard
fro
m le
ft t
o r
igh
t an
d t
he
oth
er m
ust
go
do
wn
war
d,s
o t
hei
r sl
op
es m
ust
hav
e o
pp
osi
te s
ign
s.R
ecip
roca
ls h
ave
the
sam
e si
gn
.Th
e p
rod
uct
of
the
slo
pes
mu
st b
e �
1,n
ot
1.R
emem
ber
:T
he
slo
pes
are
op
po
site
reci
pro
cals
.
©G
lenc
oe/M
cGra
w-H
ill14
2G
lenc
oe G
eom
etry
Slo
pes
an
d P
oly
go
ns
In c
oord
inat
e ge
omet
ry,t
he
slop
es o
f tw
o li
nes
det
erm
ine
if t
he
lin
es a
repa
rall
el o
r pe
rpen
dicu
lar.
Th
is k
now
ledg
e ca
n b
e u
sefu
l w
hen
wor
kin
g w
ith
poly
gon
s.
1.T
he
coor
din
ates
of
the
vert
ices
of
a tr
ian
gle
are
A(�
6,4)
,B(8
,6),
and
C(4
,�4)
.Gra
ph �
AB
C.
2.J,
K,a
nd
Lar
e m
idpo
ints
of
A�B�
,B�C�
,an
d A�
C�,
resp
ecti
vely
.Fin
d th
e co
ordi
nat
es o
f J,
K,a
nd
L.
Dra
w �
JK
L.
J(1,
5),K
(6,1
),L
(�1,
0)
3.W
hic
h s
egm
ents
app
ear
to b
e pa
rall
el?
A�B�
and
L�K�
,B�C�
and
J�L�,
A�C�
and
J�K�
4.S
how
th
at t
he
segm
ents
nam
ed i
n E
xerc
ise
3 ar
e pa
rall
el b
y fi
ndi
ng
the
slop
es o
f al
l si
x se
gmen
ts.
A�B�
:�1 7� ;
L�K�:
�1 7� ;B�
C�:�
5 2� ;J�L�
:�5 2� ;
A�C�
:�
�4 5� ;J�K�
:��4 5�
Th
e co
ord
inat
es o
f th
e ve
rtic
es o
f ri
ght
�P
QR
are
give
n.F
ind
th
e sl
ope
of e
ach
sid
e of
th
e tr
ian
gle.
Th
en n
ame
the
hyp
oten
use
.
5.P
(5,1
)Q
(1,�
1)R
(�2,
5)6.
P(�
2,�
3)Q
(5,1
)R
(2,3
)
slop
e of
P�Q�
��1 2�
slop
e of
P�Q�
��4 7�
slop
e of
Q�R�
��
2sl
ope
of Q�
R��
��2 3�
slop
e of
P�R�
��
�4 7�sl
ope
of P�
R��
�3 2�
hyp
oten
use
:P�R�
hyp
oten
use
:P�Q�
Th
e co
ord
inat
es o
f q
uad
rila
tera
l P
QR
Sar
e gi
ven
.Gra
ph
qu
adri
late
ral
PQ
RS
and
fin
d t
he
slop
es o
f th
e d
iago
nal
s.S
tate
wh
eth
er t
he
dia
gon
als
are
per
pen
dic
ula
r.
7.P
(�2,
6),Q
(4,0
),R
(1,�
4),S
(�5,
2)8.
P(0
,6),
Q(3
,0),
R(�
4,�
2),S
(�5,
4)
P�R�
:�
�1 30 �;
S�Q�
:�
�2 9� ;n
oP�
R�:
2;S�
Q�:
��1 2� ;
yes
S
P
Q
R
x
y
O
S
P
Q
R
x
y
O
A
B
C
K
J
Lx
y
O
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-3
3-3
Answers (Lesson 3-3)
© Glencoe/McGraw-Hill A11 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Eq
uat
ion
s o
f L
ines
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-4
3-4
©G
lenc
oe/M
cGra
w-H
ill14
3G
lenc
oe G
eom
etry
Lesson 3-4
Wri
te E
qu
atio
ns
of
Lin
esYo
u c
an w
rite
an
equ
atio
n o
f a
lin
e if
you
are
giv
en a
ny
ofth
e fo
llow
ing:
•th
e sl
ope
and
the
y-in
terc
ept,
•th
e sl
ope
and
the
coor
din
ates
of
a po
int
on t
he
lin
e,or
•th
e co
ordi
nat
es o
f tw
o po
ints
on
th
e li
ne.
If m
is t
he
slop
e of
a l
ine,
bis
its
y-i
nte
rcep
t,an
d (x
1,y 1
) is
a p
oin
t on
th
e li
ne,
then
:•
the
slop
e-in
terc
ept
form
of t
he
equ
atio
n i
s y
�m
x�
b,•
the
poi
nt-
slop
e fo
rmof
th
e eq
uat
ion
is
y�
y 1�
m(x
�x 1
).
Wri
te a
n e
qu
atio
n i
nsl
ope-
inte
rcep
t fo
rm o
f th
e li
ne
wit
hsl
ope
�2
and
y-i
nte
rcep
t 4.
y�
mx
�b
Slo
pe-in
terc
ept
form
y�
�2x
�4
m�
�2,
b�
4
Th
e sl
ope-
inte
rcep
t fo
rm o
f th
e eq
uat
ion
of
the
lin
e is
y�
�2x
�4.
Wri
te a
n e
qu
atio
n i
np
oin
t-sl
ope
form
of
the
lin
e w
ith
slo
pe
��3 4�
that
con
tain
s (8
,1).
y�
y 1�
m(x
�x 1
)P
oint
-slo
pe f
orm
y�
1 �
��3 4� (
x�
8)m
��
�3 4� , (
x 1,
y 1)
�(8
, 1)
Th
e po
int-
slop
e fo
rm o
f th
e eq
uat
ion
of
the
lin
e is
y�
1 �
��3 4�
(x�
8).
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Wri
te a
n e
qu
atio
n i
n s
lop
e-in
terc
ept
form
of
the
lin
e h
avin
g th
e gi
ven
slo
pe
and
y-
inte
rcep
t.
1.m
:2,y
-in
terc
ept:
�3
2.m
:��1 2� ,
y-in
terc
ept:
4
y�
2x�
3y
��
�1 2� x�
4
3.m
:�1 4� ,
y-in
terc
ept:
54.
m:0
,y-i
nte
rcep
t:�
2
y�
�1 4� x�
5y
��
2
5.m
:��5 3� ,
y-in
terc
ept:
�1 3�6.
m:�
3,y-
inte
rcep
t:�
8
y�
��5 3� x
��1 3�
y�
�3x
�8
Wri
te a
n e
qu
atio
n i
n p
oin
t-sl
ope
form
of
the
lin
e h
avin
g th
e gi
ven
slo
pe
that
con
tain
s th
e gi
ven
poi
nt.
7.m
��1 2� ,
(3,�
1)8.
m�
�2,
(4,�
2)
y�
1 �
�1 2� (x
�3)
y�
2 �
�2(
x�
4)
9.m
��
1,(�
1,3)
10.m
��1 4� ,
(�3,
�2)
y�
3 �
�(x
�1)
y�
2 �
�1 4� (x
�3)
11.m
��
�5 2� ,(0
,�3)
12.m
�0,
(�2,
5)
y�
3 �
��5 2� x
y�
5 �
0
©G
lenc
oe/M
cGra
w-H
ill14
4G
lenc
oe G
eom
etry
Wri
te E
qu
atio
ns
to S
olv
e Pr
ob
lem
sM
any
real
-wor
ld s
itu
atio
ns
can
be
mod
eled
usi
ng
lin
ear
equ
atio
ns.
Don
na
offe
rs c
omp
ute
r se
rvic
es t
o sm
all
com
pan
ies
in h
er c
ity.
Sh
ech
arge
s $5
5 p
er m
onth
for
mai
nta
inin
g a
web
sit
e an
d $
45 p
er h
our
for
each
serv
ice
call
.Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Eq
uat
ion
s o
f L
ines
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-4
3-4
Exam
ple
Exam
ple
a.W
rite
an
eq
uat
ion
to
rep
rese
nt
the
tota
lm
onth
ly c
ost
Cfo
rm
ain
tain
ing
a w
eb s
ite
and
for
hh
ours
of
serv
ice
call
s.
For
eac
h h
our,
the
cost
incr
ease
s $4
5.S
o th
e ra
te o
fch
ange
,or
slop
e,is
45.
Th
ey-
inte
rcep
t is
loc
ated
wh
ere
ther
e ar
e 0
hou
rs,o
r $5
5.C
�m
h�
b�
45h
�55
b.
Don
na
may
ch
ange
her
cos
ts t
o re
pre
sen
t th
emb
y th
e eq
uat
ion
C�
25h
�12
5,w
her
e $1
25 i
s th
efi
xed
mon
thly
fee
for
a w
eb s
ite
and
th
e co
st p
erh
our
is $
25.C
omp
are
her
new
pla
n t
o th
e ol
d o
ne
if a
com
pan
y h
as 5
�1 2�h
ours
of
serv
ice
call
s.U
nd
er
wh
ich
pla
n w
ould
Don
na
earn
mor
e?
Fir
st p
lan
For
5�1 2�
hou
rs o
f se
rvic
e D
onn
a w
ould
ear
n
C�
45h
�55
�45
�5�1 2� ��55
�24
7.5
�55
or
$302
.50
Sec
ond
Pla
n
For
5�1 2�
hou
rs o
f se
rvic
e D
onn
a w
ould
ear
n
C�
25h
�12
5 �
25(5
.5)
�12
5
�13
7.5
�12
5 or
$26
2.50
Don
na
wou
ld e
arn
mor
e w
ith
th
e fi
rst
plan
.
Exer
cises
Exer
cises
For
Exe
rcis
es 1
–4,u
se t
he
foll
owin
g in
form
atio
n.
Jerr
i’s c
urre
nt s
atel
lite
tel
evis
ion
serv
ice
char
ges
a fl
at r
ate
of $
34.9
5 pe
r m
onth
for
the
bas
icch
anne
ls a
nd a
n ad
diti
onal
$10
per
mon
th f
or e
ach
prem
ium
cha
nnel
.A c
ompe
ting
sat
elli
tete
levi
sion
ser
vice
cha
rges
a f
lat
rate
of
$39.
99 p
er m
onth
for
the
bas
ic c
hann
els
and
anad
diti
onal
$8
per
mon
th f
or e
ach
prem
ium
cha
nnel
.
1.W
rite
an
equ
atio
n i
n s
lope
-in
terc
ept
form
that
mod
els
the
tota
l m
onth
ly c
ost
for
each
sat
elli
te s
ervi
ce,w
her
e p
is t
he
nu
mbe
r of
pre
miu
m c
han
nel
s.C
urr
ent
serv
ice:
C�
10p
�34
.95
Co
mp
etin
g s
ervi
ce:C
�8p
�39
.99
3.A
th
ird
sate
llit
e co
mpa
ny
char
ges
a fl
atra
te o
f $6
9 fo
r al
l ch
ann
els,
incl
udi
ng
the
prem
ium
ch
ann
els.
If J
erri
wan
ts t
o ad
da
fou
rth
pre
miu
m c
han
nel
,wh
ich
ser
vice
wou
ld b
e le
ast
expe
nsi
ve?
the
thir
d c
om
pan
y
2.If
Jer
ri w
ants
to
incl
ude
th
ree
prem
ium
chan
nel
s in
her
pac
kage
,wh
ich
ser
vice
wou
ld b
e le
ss,h
er c
urr
ent
serv
ice
or t
he
com
peti
ng
serv
ice?
com
pet
ing
ser
vice
4.W
rite
a d
escr
ipti
on o
f h
ow t
he
fee
for
the
nu
mbe
r of
pre
miu
m c
han
nel
s is
ref
lect
edin
th
e eq
uat
ion
.T
he
fee
for
the
nu
mb
er o
f p
rem
ium
ch
ann
els
rep
rese
nts
th
e ra
te o
f ch
ang
e,o
r sl
op
e,o
f th
e eq
uat
ion
.
Answers (Lesson 3-4)
© Glencoe/McGraw-Hill A12 Glencoe Geometry
Skil
ls P
ract
ice
Eq
uat
ion
s o
f L
ines
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-4
3-4
©G
lenc
oe/M
cGra
w-H
ill14
5G
lenc
oe G
eom
etry
Lesson 3-4
Wri
te a
n e
qu
atio
n i
n s
lop
e-in
terc
ept
form
of
the
lin
e h
avin
g th
e gi
ven
slo
pe
and
y-
inte
rcep
t.
1.m
:�4,
y-in
terc
ept:
32.
m:3
,y-i
nte
rcep
t:�
8
y�
�4x
�3
y�
3x�
8
3.m
:�3 7� ,
(0,1
)4.
m:�
�2 5� ,(0
,�6)
y�
�3 7� x�
1y
��
�2 5� x�
6
Wri
te e
qu
atio
ns
in p
oin
t-sl
ope
form
an
d s
lop
e-in
terc
ept
form
of
the
lin
e h
avin
gth
e gi
ven
slo
pe
and
con
tain
ing
the
give
n p
oin
t.
5.m
:2,(
5,2)
6.m
:�3,
(2,�
4)
y�
2 �
2(x
�5)
,y�
2x�
8y
�4
��
3(x
�2)
,y�
�3x
�2
7.m
:��1 2� ,
(�2,
5)8.
m:�
1 3� ,(�
3,�
8)
y�
5 �
��1 2� (
x�
2),y
��
�1 2� x�
4y
�8
��1 3� (
x�
3),y
��1 3� x
�7
Wri
te a
n e
qu
atio
n i
n s
lop
e-in
terc
ept
form
for
eac
h l
ine.
9.r
y�
x�
310
.sy
��
2x�
2
11.t
y�
3x�
312
.uy
��1 3� x
�5
13.t
he
lin
e pa
rall
el t
o li
ne
rth
at c
onta
ins
(1,�
1)y
�x
�2
14.t
he
lin
e pe
rpen
dicu
lar
to l
ine
sth
at c
onta
ins
(0,0
)y
��1 2� x
Wri
te a
n e
qu
atio
n i
n s
lop
e-in
terc
ept
form
for
th
e li
ne
that
sat
isfi
es t
he
give
nco
nd
itio
ns.
15.m
�6,
y-in
terc
ept
��
216
.m�
��5 3� ,
y-in
terc
ept
�0
y�
6x�
2y
��
�5 3� x
17.m
��
1,co
nta
ins
(0,�
6)18
.m�
4,co
nta
ins
(2,5
)
y�
�x
�6
y�
4x�
3
19.c
onta
ins
(2,0
) a
nd
(0,1
0)20
.x-i
nte
rcep
t is
�2,
y-in
terc
ept
is �
1
y�
�5x
�10
y�
��1 2� x
�1
x
y
O
r
s
t
u
©G
lenc
oe/M
cGra
w-H
ill14
6G
lenc
oe G
eom
etry
Wri
te a
n e
qu
atio
n i
n s
lop
e-in
terc
ept
form
of
the
lin
e h
avin
g th
e gi
ven
slo
pe
and
y-
inte
rcep
t.
1.m
:�2 3� ,
y-in
terc
ept:
�10
2.m
:��7 9� ,
�0,�
�1 2� �3.
m:4
.5,(
0,0.
25)
y�
�2 3� x�
10y
��
�7 9� x�
�1 2�y
�4.
5x�
0.25
Wri
te e
qu
atio
ns
in p
oin
t-sl
ope
form
an
d s
lop
e-in
terc
ept
form
of
the
lin
e h
avin
gth
e gi
ven
slo
pe
and
con
tain
ing
the
give
n p
oin
t.
4.m
:�3 2� ,
(4,6
)5.
m:�
�6 5� ,(�
5,�
2)
y�
6 �
�3 2� (x
�4)
,y�
�3 2� xy
�2
��
�6 5� (x
�5)
,y�
��6 5� x
�8
6.m
:0.5
,(7,
�3)
7.m
:�1.
3,(�
4,4)
y�
3 �
0.5(
x�
7),
y�
4 �
�1.
3(x
�4)
,y
�0.
5x�
6.5
y�
�1.
3x�
1.2
Wri
te a
n e
qu
atio
n i
n s
lop
e-in
terc
ept
form
for
eac
h l
ine.
8.b
y�
�x
�5
9.c
y�
��2 5� x
�4
10.p
aral
lel
to l
ine
b,co
nta
ins
(3,�
2)y
��
x�
1
11.p
erpe
ndi
cula
r to
lin
e c,
con
tain
s (�
2,�
4)y
��5 2� x
�1
Wri
te a
n e
qu
atio
n i
n s
lop
e-in
terc
ept
form
for
th
e li
ne
that
sat
isfi
es t
he
give
nco
nd
itio
ns.
12.m
��
�4 9� ,y-
inte
rcep
t �
213
.m�
3,co
nta
ins
(2,�
3)
y�
��4 9� x
�2
y�
3x�
9
14.x
-in
terc
ept
is �
6,y-
inte
rcep
t is
215
.x-i
nte
rcep
t is
2,y
-in
terc
ept
is �
5
y�
�1 3� x�
2y
��5 2� x
�5
16.p
asse
s th
rou
gh (
2,�
4) a
nd
(5,8
)17
.con
tain
s (�
4,2)
an
d (8
,�1)
y�
4x�
12y
��
�1 4� x�
1
18.C
OM
MU
NIT
Y E
DU
CA
TIO
NA
loc
al c
omm
un
ity
cen
ter
offe
rs s
elf-
defe
nse
cla
sses
for
teen
s.A
$25
en
roll
men
t fe
e co
vers
su
ppli
es a
nd
mat
eria
ls a
nd
open
cla
sses
cos
t $1
0ea
ch.W
rite
an
equ
atio
n t
o re
pres
ent
the
tota
l co
st o
f x
self
-def
ense
cla
sses
at
the
com
mu
nit
y ce
nte
r.C
�10
x�
25
x
y
O
c
b
Pra
ctic
e (
Ave
rag
e)
Eq
uat
ion
s o
f L
ines
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-4
3-4
Answers (Lesson 3-4)
© Glencoe/McGraw-Hill A13 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csE
qu
atio
ns
of
Lin
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-4
3-4
©G
lenc
oe/M
cGra
w-H
ill14
7G
lenc
oe G
eom
etry
Lesson 3-4
Pre-
Act
ivit
yH
ow c
an t
he
equ
atio
n o
f a
lin
e d
escr
ibe
the
cost
of
cell
ula
rte
lep
hon
e se
rvic
e?
Rea
d th
e in
trod
uct
ion
to
Les
son
3-4
at
the
top
of p
age
145
in y
our
text
book
.If
th
e ra
tes
for
you
r ce
llu
lar
phon
e pl
an a
re d
escr
ibed
by
the
equ
atio
n i
nyo
ur
text
book
,wh
at w
ill
be t
he
tota
l ch
arge
(ex
clu
din
g ta
xes
and
fees
) fo
r a
mon
th i
n w
hic
h y
ou u
se 5
0 m
inu
tes
of a
ir t
ime?
$23
.45
Rea
din
g t
he
Less
on
1.Id
enti
fy w
hat
eac
h f
orm
ula
rep
rese
nts
.
a.y
�y 1
�m
(x�
x 1)
po
int-
slo
pe
form
of
an e
qu
atio
n
b.
m�
�y x2 2
� �
y x1 1�
slo
pe
of
a lin
e
c.y
�m
x�
bsl
op
e-in
terc
ept
form
of
an e
qu
atio
n
2.W
rite
th
e po
int-
slop
e fo
rm o
f th
e eq
uat
ion
for
eac
h l
ine.
a.li
ne
wit
h s
lope
��1 2�
con
tain
ing
(�2,
5)y
�5
��
�1 2� (x
�2)
b.
lin
e co
nta
inin
g (�
4.5,
�6.
5) a
nd
para
llel
to
a li
ne
wit
h s
lope
0.5
y�
6.5
�0.
5(x
�4.
5)3.
Wh
ich
on
e of
th
e fo
llow
ing
corr
ectl
y de
scri
bes
the
y-in
terc
ept
of a
lin
e?C
A.
the
y-co
ordi
nat
e of
th
e po
int
wh
ere
the
lin
e in
ters
ects
th
e x-
axis
B.t
he
x-co
ordi
nat
e of
th
e po
int
wh
ere
the
lin
e in
ters
ects
th
e y-
axis
C.
the
y-co
ordi
nat
e of
th
e po
int
wh
ere
the
lin
e cr
osse
s th
e y-
axis
D.t
he
x-co
ordi
nat
e of
th
e po
int
wh
ere
the
lin
e cr
osse
s th
e x-
axis
E.
the
rati
o of
th
e ch
ange
in
y-c
oord
inat
es t
o th
e ch
ange
in
x-c
oord
inat
es
4.F
ind
the
slop
e an
d y-
inte
rcep
t of
eac
h l
ine.
a.y
�2x
�7
slo
pe
�2;
y-in
terc
ept
��
7b
.x
�y
�8.
5sl
op
e �
�1;
y-in
terc
ept
�8.
5c.
2.4x
�y
�4.
8sl
op
e �
2.4;
y-in
terc
ept
��
4.8
d.
y�
7 �
x�
12sl
op
e �
1;y-
inte
rcep
t �
19e.
y�
5 �
�2(
x�
6)sl
op
e �
�2;
y-in
terc
ept
��
17
Hel
pin
g Y
ou
Rem
emb
er5.
A g
ood
way
to
rem
embe
r so
met
hin
g n
ew i
s to
rel
ate
it t
o so
met
hin
g yo
u a
lrea
dy k
now
.H
ow c
an t
he
slop
e fo
rmu
la h
elp
you
to
rem
embe
r th
e eq
uat
ion
for
th
e po
int-
slop
e fo
rm
of a
lin
e?S
amp
le a
nsw
er:T
he
slo
pe
of
a lin
e th
rou
gh
(x,
y)
and
(x 1
,y1)
is
giv
en b
y �y x
� �
y x1 1�
�m
.Mu
ltip
ly e
ach
sid
e o
f th
e sl
op
e fo
rmu
la b
y x 2
�x 1
.
Th
e re
sult
will
be
the
po
int-
slo
pe
form
.
©G
lenc
oe/M
cGra
w-H
ill14
8G
lenc
oe G
eom
etry
Ab
solu
te Z
ero
All
mat
ter
is m
ade
up
of a
tom
s an
d m
olec
ule
s th
at a
re i
n c
onst
ant
mot
ion
.Tem
pera
ture
is
one
mea
sure
of
this
mot
ion
.Abs
olu
te z
ero
is t
he
theo
reti
cal
tem
pera
ture
lim
it a
t w
hic
h t
he
mot
ion
of
the
mol
ecu
les
and
atom
s of
a s
ubs
tan
ce i
s th
e le
ast
poss
ible
.
Exp
erim
ents
wit
h g
aseo
us
subs
tan
ces
yiel
d da
ta t
hat
all
ow y
ou t
oes
tim
ate
just
how
col
d ab
solu
te z
ero
is.F
or a
ny
gas
of a
con
stan
tvo
lum
e,th
e pr
essu
re,e
xpre
ssed
in
a u
nit
cal
led
atm
osph
eres
,va
ries
lin
earl
y as
th
e te
mpe
ratu
re.T
hat
is,
the
pres
sure
Pan
d th
ete
mpe
ratu
re t
are
rela
ted
by a
n e
quat
ion
of
the
form
P�
mt�
b,w
her
e m
and
bar
e re
al n
um
bers
.
1.S
ketc
h a
gra
ph f
or t
he
data
in
th
e ta
ble.
2.U
se t
he
data
an
d yo
ur
grap
h t
o fi
nd
valu
es f
or m
an
d b
in t
he
equ
atio
nP
�m
t�b,
wh
ich
rel
ates
tem
pera
ture
to
pres
sure
.
m�
0.00
36,b
�1.
00
3.E
stim
ate
abso
lute
zer
o in
deg
rees
Cel
siu
s by
set
tin
g P
equ
al t
o 0
in
the
equ
atio
n a
bove
an
d u
sin
g th
e va
lues
man
d b
that
you
obt
ain
ed
in E
xerc
ise
2.
abo
ut
�27
8�C
(T
he
actu
al v
alu
e is
slig
htl
y h
igh
er.)
25–2
550
7510
012
5
0.5
1.5
1.0
t
P
O
tP
(in
�C
)(i
n a
tmo
sph
eres
)
�25
0.91
01.
00
251.
09
100
1.36
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-4
3-4
Answers (Lesson 3-4)
© Glencoe/McGraw-Hill A14 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Pro
vin
g L
ines
Par
alle
l
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-5
3-5
©G
lenc
oe/M
cGra
w-H
ill14
9G
lenc
oe G
eom
etry
Lesson 3-5
Iden
tify
Par
alle
l Lin
esIf
tw
o li
nes
in
a p
lan
e ar
e cu
t by
a t
ran
sver
sal
and
cert
ain
con
diti
ons
are
met
,th
en t
he
lin
es m
ust
be
para
llel
.
Ifth
en
•co
rres
pond
ing
angl
es a
re c
ongr
uent
,th
e lin
es a
re p
aral
lel.
•al
tern
ate
exte
rior
angl
es a
re c
ongr
uent
,•
cons
ecut
ive
inte
rior
angl
es a
re s
uppl
emen
tary
,•
alte
rnat
e in
terio
r an
gles
are
con
grue
nt,
or•
two
lines
are
per
pend
icul
ar t
o th
e sa
me
line,
If m
�1
�m
�2,
det
erm
ine
wh
ich
lin
es,i
f an
y,ar
ep
aral
lel.
Sin
ce m
�1
�m
�2,
then
�1
��
2.�
1 an
d�
2 ar
e co
ngr
uen
t co
rres
pon
din
g an
gles
,so
r|| s
.
nm
rs
12
Fin
d x
and
m�
AB
Cso
th
at m
|| n.
We
can
con
clu
de t
hat
m|| n
if a
lter
nat
ein
teri
or a
ngl
es a
re c
ongr
uen
t.
m�
DA
B�
m�
CD
A3x
�10
�6x
�20
10 �
3x�
2030
�3x
10 �
x
m�
AB
C�
6x�
20�
6(10
) �
20 o
r 40
n
mA
B
C
D
( 3x
� 1
0)�
( 6x
� 2
0)�
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d x
so t
hat
�|| m
.
1.2.
3.
1510
25
4.5.
6.
720
10
m n
�
( 5x
� 2
0)�
70�
m� ( 3
x �
20)
�
2 x�
m�
( 9x
� 1
) �( 8x
� 8
) �
m� ( 3
x �
15)
�
m�
( 4x
� 2
0)�
6 x�
m�
( 5x
� 5
) �
( 6x
� 2
0)�
©G
lenc
oe/M
cGra
w-H
ill15
0G
lenc
oe G
eom
etry
Pro
ve L
ines
Par
alle
lYo
u c
an p
rove
th
at l
ines
are
par
alle
l by
usi
ng
pos
tula
tes
and
theo
rem
s ab
out
pair
s of
an
gles
.You
als
o ca
n u
se s
lope
s of
lin
es t
o pr
ove
that
tw
o li
nes
are
para
llel
or
perp
endi
cula
r.
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Pro
vin
g L
ines
Par
alle
l
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-5
3-5
Exam
ple
Exam
ple
For
Exe
rcis
es 1
–6,f
ill
in t
he
bla
nk
s.G
iven
:�1
��
5,�
15 �
�5
Pro
ve: �
|| m,r
|| s
Sta
tem
ents
Rea
son
s
1.�
15 �
�5
1.G
iven
2.�
13 �
�15
2.V
erti
cal �
are
�.
3.�
5 �
�13
3.Tr
ansi
tive
Pro
per
ty o
f �
4.r
|| s4.
If c
orr
.�ar
e �
,th
en li
nes
|| .5.
�1
��
55.
Giv
en
6.�
|| m6.
If c
orr
�ar
e �
,th
en l
ines
|| .
7.D
eter
min
e w
het
her
PQ
���
⊥T
Q��
� .E
xpla
in w
hy
or w
hy
not
.
slo
pe
of
PQ
��� :
��1 2� ;
slo
pe
of
TQ��
� :2
���1 2� ��
2 �
�1,
so P
Q��
�⊥
TQ��
� .
x
y O
PQ
T
� m
rs
12
43 5
68
7
910 11
12
1413
1516
aG
iven
:�1
��
2,�
1 �
�3
Pro
ve:A�
B�|| D�
C�
Sta
tem
ents
Rea
son
s1.
�1
��
21.
Giv
en�
1 �
�3
2.�
2 �
�3
2.T
ran
siti
ve P
rope
rty
of �
3.A �
B�|| D�
C�3.
If a
lt.i
nt.
angl
es a
re �
,th
enth
e li
nes
are
|| .
123
AB
CD
b.
Wh
ich
lin
es a
re p
aral
lel?
Wh
ich
lin
es a
re p
erp
end
icu
lar?
slop
e of
P�Q�
�0
slop
e of
S�R�
�0
slop
e of
P�S�
��4 3�
slop
e of
Q�R�
��4 3�
slop
e of
P�R�
��
2sl
ope
of S�
Q��
�1 2�
So
P�Q�
|| S�R�
,P�S�
|| Q�R�
,an
d P�
R�⊥
S�Q�
.
x
y
O4
8–4
–8
8 4 –4 –8
P( –
2, 4
)Q
( 8, 4
)
S( –
8, –
4)R
( 2, –
4)
Exer
cises
Exer
cises
Answers (Lesson 3-5)
© Glencoe/McGraw-Hill A15 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Pro
vin
g L
ines
Par
alle
l
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-5
3-5
©G
lenc
oe/M
cGra
w-H
ill15
1G
lenc
oe G
eom
etry
Lesson 3-5
Giv
en t
he
foll
owin
g in
form
atio
n,d
eter
min
e w
hic
h l
ines
,if
an
y,ar
e p
aral
lel.
Sta
te t
he
pos
tula
te o
r th
eore
m t
hat
ju
stif
ies
you
r an
swer
.
1.�
3 �
�7
2.�
9 �
�11
a|| b
;al
t.in
t.�
a|| b
;co
rr.�
3.�
2 �
�16
4.m
�5
�m
�12
�18
0
�|| m
;al
t.ex
t.�
�|| m
;co
ns.
int.
�
Fin
d x
so t
hat
�|| m
.
5.22
6.20
7.6
8.PR
OO
FP
rovi
de a
rea
son
for
eac
h s
tate
men
t in
th
e pr
oof
of T
heo
rem
3.7
.G
iven
:�1
and
�2
are
com
plem
enta
ry.
B �C�
⊥C�
D�P
rove
:B�A�
|| C�D�
Pro
of:
Sta
tem
ents
Rea
son
s1.
B �C�
⊥C�
D�1.
Giv
en2.
m�
AB
C�
m�
1 �
m�
22.
An
gle
Ad
dit
ion
Po
stu
late
3.�
1 an
d �
2 ar
e co
mpl
emen
tary
.3.
Giv
en4.
m�
1 �
m�
2 �
904.
Def
init
ion
of
com
ple
men
tary
an
gle
s5.
m�
AB
C�
905.
Tran
siti
ve P
rop
erty
of
Eq
ual
ity
6.B�
A�⊥
B�C�
6.D
efin
itio
n o
f p
erp
end
icu
lar
7.B�
A�|| C�
D�7.
If 2
lin
es a
re ⊥
to t
he
sam
e lin
e,th
enlin
es a
re || .
Det
erm
ine
wh
eth
er e
ach
pai
r of
lin
es i
s p
aral
lel.
Exp
lain
wh
y or
wh
y n
ot.
9.10
.
Yes;
the
slo
pes
are
th
e sa
me.
No
;th
e sl
op
es a
re n
ot
the
sam
e.
x
y
O
U( 4
, 2)
T( –
4, 0
)
S( 5
, –3)
R( 0
, –4)
x
y O
A( 1
, 3)
B( –
2, –
3)E
( 0, –
3)F( 2
, 1)
1
ADC
B2
m
k�
( 6x
� 4
) � ( 8x
� 8
) �m
k
�
( 4x
� 1
0)�
( 3x
� 1
0)�
m
k�
( 2x
� 6
) �
130�
m�
ab
12
34
87
65
910
1112
1615
1413
©G
lenc
oe/M
cGra
w-H
ill15
2G
lenc
oe G
eom
etry
Giv
en t
he
foll
owin
g in
form
atio
n,d
eter
min
e w
hic
h l
ines
,if
an
y,ar
e p
aral
lel.
Sta
te t
he
pos
tula
te o
r th
eore
m t
hat
ju
stif
ies
you
r an
swer
.
1.m
�B
CG
�m
�F
GC
�18
02.
�C
BF
��
GF
H
BD
���
|| EG
��� ;
BD
���
|| EG
��� ;
con
s.in
t.�
corr
.�
3.�
EF
B�
�F
BC
4.�
AC
D�
�K
BF
BD
���
|| EG
��� ;
AJ
���
|| BH
��� ;
alt.
int.
�al
t.ex
t.�
Fin
d x
so t
hat
�|| m
.
5.12
6.21
7.9
8.PR
OO
FW
rite
a t
wo-
colu
mn
pro
of.
Giv
en:�
2 an
d �
3 ar
e su
pple
men
tary
.P
rove
:A �B�
|| C�D�
Pro
of:
Sta
tem
ents
Rea
son
s
1.�
2 an
d �
3 ar
e su
pp
lem
enta
ry.
1.G
iven
2.A
B��
�|| C
D��
�2.
If c
on
sec.
int
�ar
e su
pp
l.,th
en li
nes
are
|| .3.
A�B�
|| C�D�
3.S
egm
ents
co
nta
ined
in ||
lines
are
|| .
9.LA
ND
SCA
PIN
GT
he
hea
d ga
rden
er a
t a
bota
nic
al g
arde
n w
ants
to
plan
t ro
sebu
shes
in
para
llel
row
s on
eit
her
sid
e of
an
exi
stin
g fo
otpa
th.H
ow c
an t
he
gard
ener
en
sure
th
atth
e ro
ws
are
para
llel
?
Sam
ple
an
swer
:If
th
e g
ard
ener
dig
s ea
ch r
ow
at
a 90
�an
gle
to
th
efo
otp
ath
,eac
h r
ow
will
be
per
pen
dic
ula
r to
th
e fo
otp
ath
.If
each
of
the
row
s is
per
pen
dic
ula
r to
th
e fo
otp
ath
,th
en t
he
row
s ar
e p
aral
lel.1
42
5
36
AB
CD
( 2x
� 1
2)� ( 5x
� 1
5)�
t
m �
( 5x
� 1
8)�
( 7x
� 2
4)�
t m
�
( 3x
� 6
) �
( 4x
� 6
) �
t m
�
A
BC
D
EF H
K
G J
Pra
ctic
e (
Ave
rag
e)
Pro
vin
g L
ines
Par
alle
l
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-5
3-5
Answers (Lesson 3-5)
© Glencoe/McGraw-Hill A16 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csP
rovi
ng
Lin
es P
aral
lel
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-5
3-5
©G
lenc
oe/M
cGra
w-H
ill15
3G
lenc
oe G
eom
etry
Lesson 3-5
Pre-
Act
ivit
yH
ow d
o yo
u k
now
th
at t
he
sid
es o
f a
par
kin
g sp
ace
are
par
alle
l?
Rea
d th
e in
trod
uct
ion
to
Les
son
3-5
at
the
top
of p
age
151
in y
our
text
book
.
How
can
th
e w
orke
rs w
ho
are
stri
pin
g th
e pa
rkin
g sp
aces
in
a p
arki
ng
lot
chec
k to
see
if
the
side
s of
th
e sp
aces
are
par
alle
l?S
amp
le a
nsw
er:
Use
a T-
squ
are
or
oth
er d
evic
e fo
r fo
rmin
g r
igh
t an
gle
s to
lay
ou
tp
erp
end
icu
lar
seg
men
ts c
ut
fro
m s
trin
g o
r ro
pe
con
nec
tin
gth
e tw
o s
ides
of
a p
arki
ng
sp
ace
at b
oth
en
ds
and
in t
he
mid
dle
.Mea
sure
th
e th
ree
per
pen
dic
ula
r se
gm
ents
.If
they
are
all t
he
sam
e le
ng
th,t
he
sid
es o
f th
e p
arki
ng
sp
ace
are
par
alle
l.
Rea
din
g t
he
Less
on
1.C
hoo
se t
he
wor
d or
ph
rase
th
at b
est
com
plet
es e
ach
sen
ten
ce.
a.If
tw
o co
plan
ar l
ines
are
cu
t by
a t
ran
sver
sal
so t
hat
cor
resp
ondi
ng
angl
es a
re
con
gru
ent,
then
th
e li
nes
are
(p
aral
lel/p
erpe
ndi
cula
r/sk
ew).
b.
In a
pla
ne,
if t
wo
lin
es a
re p
erpe
ndi
cula
r to
th
e sa
me
lin
e,th
en t
hey
are
(per
pen
dicu
lar/
para
llel
/ske
w).
c.F
or a
lin
e an
d a
poin
t n
ot o
n t
he
lin
e,th
ere
exis
ts
(at
leas
t on
e/ex
actl
y on
e/at
mos
t on
e) l
ine
thro
ugh
th
e po
int
that
is
para
llel
to
the
give
n l
ine.
d.
If t
wo
copl
anar
lin
es a
re c
ut
by a
tra
nsv
ersa
l so
th
at c
onse
cuti
ve i
nte
rior
an
gles
are
(com
plem
enta
ry/s
upp
lem
enta
ry/c
ongr
uen
t),t
hen
th
e li
nes
are
para
llel
.
e.If
tw
o co
plan
ar l
ines
are
cu
t by
a t
ran
sver
sal
so t
hat
alt
ern
ate
inte
rior
an
gles
are
con
gru
ent,
then
th
e li
nes
are
(p
erpe
ndi
cula
r/pa
rall
el/s
kew
).
2.W
hic
h o
f th
e fo
llow
ing
con
diti
ons
veri
fy t
hat
p|| q
?A
,C,F
,GA
.�
6 �
�12
B.�
2 �
�4
C.
�8
��
16D
.�11
��
13
E.
�6
and
�7
are
supp
lem
enta
ry.
F.�
1 �
�15
G.�
7 an
d �
10 a
re s
upp
lem
enta
ry.
H.�
4 �
�16
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
som
eth
ing
new
is
to d
raw
a p
ictu
re.H
ow c
an a
ske
tch
hel
p yo
uto
rem
embe
r th
e P
aral
lel
Pos
tula
te?
Sam
ple
an
swer
:D
raw
a li
ne
wit
h a
ru
ler
or
stra
igh
ted
ge
and
ch
oo
se a
po
int
no
t o
n t
he
line.
Try
to d
raw
lin
es t
hro
ug
h t
he
po
int
that
are
par
alle
lto
th
e lin
e yo
u o
rig
inal
ly d
rew
.Yo
u w
ill s
ee t
hat
th
ere
is e
xact
ly o
ne
way
to d
o t
his
.
p
t
q r
12 9
10
1213
14
1516
11
34 6
5
78
par
alle
l
sup
ple
men
tary
exac
tly
on
ep
aral
lel
par
alle
l
©G
lenc
oe/M
cGra
w-H
ill15
4G
lenc
oe G
eom
etry
Scr
ambl
ed-U
p P
roo
f
Th
e re
ason
s n
eces
sary
to
com
ple
te t
he
foll
owin
g p
roof
are
scra
mb
led
up
bel
ow.T
o co
mp
lete
th
e p
roof
,nu
mb
er t
he
reas
ons
to m
atch
th
e co
rres
pon
din
g st
atem
ents
.
Giv
en:C �
D�⊥
B�E�
A�B�
⊥B�
E�A�
D��
C�E�
B�D�
�D�
E�
Pro
ve:A�
D�|| C�
E�
Pro
of:
Sta
tem
ents
Rea
son
s
1.C�
D�⊥
B�E�
Def
init
ion
of
Rig
ht
Tri
angl
e4
2.A
B⊥
B�E�
Giv
en1
3.�
3 an
d �
4 ar
e ri
ght
angl
es.
Giv
en2
4.�
AB
Dan
d �
CD
Ear
e ri
ght
tria
ngl
es.
Def
init
ion
of
Per
pen
dicu
lar
Lin
es3
5.A�
D��
C�E�
Giv
en5
6.B�
D��
D�E�
CP
CT
C8
7.�
AB
D�
�C
DE
In a
pla
ne,
if t
wo
lin
es a
re c
ut
by a
tra
nsv
ersa
lso
th
at a
pai
r of
cor
resp
ondi
ng
angl
es i
sco
ngr
uen
t,th
en t
he
lin
es a
re p
aral
lel.
(Th
eore
m 7
-5)
9
8.�
1 �
�2
Giv
en6
9.A�
D�|| C�
E�H
L7
AC
BE
D
5 31
42
6
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-5
3-5
Answers (Lesson 3-5)
© Glencoe/McGraw-Hill A17 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Per
pen
dic
ula
rs a
nd
Dis
tan
ce
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-6
3-6
©G
lenc
oe/M
cGra
w-H
ill15
5G
lenc
oe G
eom
etry
Lesson 3-6
Dis
tan
ce F
rom
a P
oin
t to
a L
ine
Wh
en a
poi
nt
is
not
on
a l
ine,
the
dist
ance
fro
m t
he
poin
t to
th
e li
ne
is t
he
len
gth
of
the
segm
ent
that
con
tain
s th
e po
int
and
ispe
rpen
dicu
lar
to t
he
lin
e.
Dra
w t
he
segm
ent
that
rep
rese
nts
th
e d
ista
nce
from
Eto
AF
��� .
Ext
end
AF
� �� .
Dra
w E�
G�⊥
AF
��� .
E �G�
repr
esen
ts t
he
dist
ance
fro
m E
to A
F��
� .
Dra
w t
he
segm
ent
that
rep
rese
nts
th
e d
ista
nce
in
dic
ated
.
1.C
to A
B� �
�2.
Dto
AB
���
3.T
to R
S��
�4.
Sto
PQ
���
5.S
to Q
R��
�6.
Sto
RT
���
XR
T
S
R
T
S
P
Q
X
RT
S
PQ
X
URS
T
X
A
CD
BX
A
C
BX
AF
G
BE
AF
BE
Pdi
stan
ce b
etw
een
M a
nd P
Q��
�
Q
M
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill15
6G
lenc
oe G
eom
etry
Dis
tan
ce B
etw
een
Par
alle
l Lin
esT
he
dist
ance
bet
wee
n p
aral
lel
lin
es i
s th
e le
ngt
hof
a s
egm
ent
that
has
an
en
dpoi
nt
on e
ach
lin
e an
d is
per
pen
dicu
lar
to t
hem
.Par
alle
l li
nes
are
ever
ywh
ere
equ
idis
tan
t,w
hic
h m
ean
s th
at a
ll s
uch
per
pen
dicu
lar
segm
ents
hav
e th
esa
me
len
gth
.
Fin
d t
he
dis
tan
ce b
etw
een
th
e p
aral
lel
lin
es �
and
mw
hos
eeq
uat
ion
s ar
e y
�2x
�1
and
y�
2x�
4,re
spec
tive
ly.
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Per
pen
dic
ula
rs a
nd
Dis
tan
ce
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-6
3-6
Exam
ple
Exam
ple
Dra
w a
lin
e p
thro
ugh
(0,
1) t
hat
is
perp
endi
cula
r to
�an
d m
.
Lin
e p
has
slo
pe �
�1 2�an
d y-
inte
rcep
t 1.
An
equ
atio
n o
f p
is y
��
�1 2� x�
1.T
he
poin
t of
inte
rsec
tion
for
pan
d �
is (
0,1)
.
x
y O
mp
�
( 0, 1
)
x
y O
m�
To
fin
d th
e po
int
of i
nte
rsec
tion
of
pan
d m
,so
lve
a sy
stem
of
equ
atio
ns.
Lin
e m
:y
�2x
�4
Lin
e p:
y�
��1 2� x
�1
Use
su
bsti
tuti
on.
2x�
4 �
��1 2� x
�1
4x�
8 �
�x
�2
5x�
10x
�2
Su
bsti
tute
2 f
or x
to f
ind
the
y-co
ordi
nat
e.
y�
��1 2� x
�1
��
�1 2� (2)
�1
��
1 �
1 �
0
Th
e po
int
of i
nte
rsec
tion
of
pan
d m
is (
2,0)
.
Use
th
e D
ista
nce
For
mu
la t
o fi
nd
the
dist
ance
bet
wee
n (
0,1)
an
d (2
,0).
d�
�(x
2�
�x 1
)2�
�(y
2�
�y 1
)2�
��
(2 �
0�
)2�
(�
0 �
1�
)2 ��
�5�
Th
e di
stan
ce b
etw
een
�an
d m
is �
5�u
nit
s.
Exer
cises
Exer
cises
Fin
d t
he
dis
tan
ce b
etw
een
eac
h p
air
of p
aral
lel
lin
es.
1.y
�8
2.y
�x
�3
3.y
��
2xy
��
3y
�x
�1
y�
�2x
�5
11�
8��
5�
Answers (Lesson 3-6)
© Glencoe/McGraw-Hill A18 Glencoe Geometry
Skil
ls P
ract
ice
Per
pen
dic
ula
rs a
nd
Dis
tan
ce
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-6
3-6
©G
lenc
oe/M
cGra
w-H
ill15
7G
lenc
oe G
eom
etry
Lesson 3-6
Dra
w t
he
segm
ent
that
rep
rese
nts
th
e d
ista
nce
in
dic
ated
.
1.B
to A
C��
�2.
Gto
EF
���
3.Q
to S
R��
�
Con
stru
ct a
lin
e p
erp
end
icu
lar
to �
thro
ugh
K.T
hen
fin
d t
he
dis
tan
ce f
rom
Kto
�.
4.5.
�13�
3�2�
Fin
d t
he
dis
tan
ce b
etw
een
eac
h p
air
of p
aral
lel
lin
es.
6.y
�7
7.x
��
68.
y�
3xy
��
1x
�5
y�
3x�
10
811
�10�
9.y
��
5x10
.y�
x�
911
.y�
�2x
�5
y�
�5x
�26
y�
x�
3y
��
2x�
5
�26�
3�2�
2�5�x
y O
�
K
x
y
O
�
K
S
PQ
RD
EF
GA
B
C
©G
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oe/M
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ill15
8G
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oe G
eom
etry
Dra
w t
he
segm
ent
that
rep
rese
nts
th
e d
ista
nce
in
dic
ated
.
1.O
to M
N��
�2.
Ato
DC
���
3.T
to V
U��
�
Con
stru
ct a
lin
e p
erp
end
icu
lar
to �
thro
ugh
B.T
hen
fin
d t
he
dis
tan
ce f
rom
Bto
�.
4.5.
4�2�
�13�
Fin
d t
he
dis
tan
ce b
etw
een
eac
h p
air
of p
aral
lel
lin
es.
6.y
��
x7.
y�
2x�
78.
y�
3x�
12y
��
x�
4y
�2x
�3
y�
3x�
18
2�2�
2�5�
3�10�
9.G
raph
th
e li
ne
y�
�x
�1.
Con
stru
ct a
per
pen
dicu
lar
segm
ent
thro
ugh
th
e po
int
at (
�2,
�3)
.Th
en f
ind
the
dist
ance
fro
m t
he
poin
t to
th
e li
ne.
3 �2 �
10.C
AN
OEI
NG
Bro
nso
n a
nd
a fr
ien
d ar
e go
ing
to c
arry
a c
anoe
acr
oss
a fl
at f
ield
to
the
ban
k of
a s
trai
ght
can
al.D
escr
ibe
the
shor
test
pat
h t
hey
can
use
.
Sam
ple
an
swer
:Th
e sh
ort
est
pat
h w
ou
ld b
e a
per
pen
dic
ula
r se
gm
ent
fro
m w
her
e th
ey a
re t
o t
he
ban
k o
f th
e ca
nal
.
( –2,
–3)
y �
�x
� 1
x
y
O
x
y O
�B
x
y O
� B
T
VW
SU
AB
CD
MN
O
Pra
ctic
e (
Ave
rag
e)
Per
pen
dic
ula
rs a
nd
Dis
tan
ce
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-6
3-6
Answers (Lesson 3-6)
© Glencoe/McGraw-Hill A19 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csP
erp
end
icu
lars
an
d D
ista
nce
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-6
3-6
©G
lenc
oe/M
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w-H
ill15
9G
lenc
oe G
eom
etry
Lesson 3-6
Pre-
Act
ivit
yH
ow d
oes
the
dis
tan
ce b
etw
een
par
alle
l li
nes
rel
ate
to h
angi
ng
new
sh
elve
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
3-6
at
the
top
of p
age
159
in y
our
text
book
.
Nam
e th
ree
exam
ples
of
situ
atio
ns
in h
ome
con
stru
ctio
n w
her
e it
wou
ld b
eim
port
ant
to c
onst
ruct
par
alle
l li
nes
.S
amp
le a
nsw
er:
op
po
site
wal
ls o
f a
roo
m,p
lan
ks o
f h
ard
wo
od
flo
ori
ng
,to
ps
and
bo
tto
ms
of
cab
inet
s
Rea
din
g t
he
Less
on
1.F
ill
in t
he
blan
k w
ith
a w
ord
or p
hra
se t
o co
mpl
ete
each
sen
ten
ce.
a.T
he
dist
ance
fro
m a
lin
e to
a p
oin
t n
ot o
n t
he
lin
e is
th
e le
ngt
h o
f th
e se
gmen
t
to t
he
lin
e fr
om t
he
poin
t.
b.
Tw
o co
plan
ar l
ines
are
par
alle
l if
th
ey a
re e
very
wh
ere
.
c.In
a p
lan
e,if
tw
o li
nes
are
bot
h e
quid
ista
nt
from
a t
hir
d li
ne,
then
th
e tw
o li
nes
are
to e
ach
oth
er.
d.
Th
e di
stan
ce b
etw
een
tw
o pa
rall
el l
ines
mea
sure
d al
ong
a pe
rpen
dicu
lar
to t
he
two
lin
es i
s al
way
s .
e.T
o m
easu
re t
he
dist
ance
bet
wee
n t
wo
para
llel
lin
es,m
easu
re t
he
dist
ance
bet
wee
n
one
of t
he
lin
es a
nd
any
poin
t on
th
e .
2.O
n e
ach
fig
ure
,dra
w t
he
segm
ent
that
rep
rese
nts
th
e di
stan
ce i
ndi
cate
d.
a.P
to �
b.D
to A
B��
�
c.E
to F
G� �
�d
.Uto
RV
� ��
Hel
pin
g Y
ou
Rem
emb
er3.
A g
ood
way
to
rem
embe
r a
new
wor
d is
to
rela
te i
t to
wor
ds t
hat
use
th
e sa
me
root
.Use
you
r di
ctio
nar
y to
fin
d th
e m
ean
ing
of t
he
Lat
in r
oot
aequ
us.
Lis
t th
ree
wor
ds o
ther
th
aneq
ual
an
d eq
uid
ista
nt
that
are
der
ived
fro
m t
his
roo
t an
d gi
ve t
he
mea
nin
g of
eac
h.
Sam
ple
an
swer
:Aeq
uus
mea
ns
even
,fai
r,o
r eq
ual.
Equ
inox
mea
ns
on
e o
fth
e tw
o t
imes
of
year
wh
en d
ay a
nd
nig
ht
are
of
equ
al le
ng
th. E
quity
mea
ns
bei
ng
just
or
fair.
Equ
ival
ent
mea
ns
bei
ng
eq
ual
in v
alu
e o
r m
ean
ing
.
TV
RS
U
E
FG
A
DC
B
P
�
oth
er li
ne
the
sam
e
par
alle
l
equ
idis
tan
tp
erp
end
icu
lar
©G
lenc
oe/M
cGra
w-H
ill16
0G
lenc
oe G
eom
etry
Par
alle
lism
in S
pac
eIn
spa
ce g
eom
etry
,th
e co
nce
pt o
f pa
rall
elis
m m
ust
be
exte
nde
d to
in
clu
de t
wo
plan
es a
nd
a li
ne
and
a pl
ane.
Def
init
ion
:T
wo
plan
es a
re p
aral
lel
if a
nd
only
if
they
do
not
in
ters
ect.
Def
init
ion
:A
lin
e an
d a
plan
e ar
e pa
rall
el i
f an
d on
ly
if t
hey
do
not
in
ters
ect.
Th
us,
in s
pace
,tw
o li
nes
can
be
inte
rsec
tin
g,pa
rall
el,o
r sk
ew w
hil
e tw
o pl
anes
or
a li
ne
and
a pl
ane
can
on
ly b
ein
ters
ecti
ng
or p
aral
lel.
In t
he
figu
re a
t th
e ri
ght,
t�M
,t�
P,P
|| H,a
nd
�an
d n
are
skew
.
Th
e fo
llow
ing
five
sta
tem
ents
are
th
eore
ms
abou
t pa
rall
el p
lan
es.
Th
eore
m:
Tw
o pl
anes
per
pen
dicu
lar
to t
he
sam
e li
ne
are
para
llel
.T
heo
rem
:T
wo
plan
es p
aral
lel
to t
he
sam
e pl
ane
are
para
llel
.T
heo
rem
:A
lin
e pe
rpen
dicu
lar
to o
ne
of t
wo
para
llel
pla
nes
is
perp
endi
cula
r to
th
e ot
her
.T
heo
rem
:A
pla
ne
perp
endi
cula
r to
on
e of
tw
o pa
rall
el p
lan
es i
s pe
rpen
dicu
lar
to t
he
oth
er.
Th
eore
m:
If t
wo
para
llel
pla
nes
eac
h i
nte
rsec
t a
thir
d pl
ane,
then
th
e tw
o li
nes
of
inte
rsec
tion
are
par
alle
l.
Use
th
e fi
gure
giv
en a
bov
e fo
r E
xerc
ises
1–1
0.S
tate
yes
or n
oto
tel
l w
het
her
th
est
atem
ent
is t
rue.
1.M
�Pye
s2.
��n
no
3.M
�Hye
s4.
��P
yes
5.�
�t
yes
6.n
�Hye
s7.
��
Pn
o8.
t�H
no
9.M
�t
yes
10.
t �H
yes
Mak
e a
smal
l sk
etch
to
show
th
at e
ach
sta
tem
ent
is f
alse
.
11.I
f tw
o li
nes
are
par
alle
l to
th
e sa
me
12.I
f tw
o pl
anes
are
par
alle
l,th
en a
ny
lin
e pl
ane,
then
th
e li
nes
are
par
alle
l.in
on
e pl
ane
is p
aral
lel
to a
ny
lin
e in
th
eot
her
pla
ne.
13.I
f tw
o li
nes
are
par
alle
l,th
en a
ny
plan
e14
.If
two
lin
es a
re p
aral
lel,
then
an
y pl
ane
con
tain
ing
one
of t
he
lin
es i
s pa
rall
el t
oco
nta
inin
g on
e of
th
e li
nes
is
para
llel
to
any
plan
e co
nta
inin
g th
e ot
her
lin
e.th
e ot
her
lin
e.
t
�M
nP
H
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-6
3-6
Answers (Lesson 3-6)
© Glencoe/McGraw-Hill A20 Glencoe Geometry
Chapter 3 Assessment Answer Key Form 1 Form 2APage 161 Page 162 Page 163
(continued on the next page)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
C
D
B
A
D
B
C
D
C
A
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
A
D
B
C
C
A
B
D
A
C
undefined
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
C
D
C
D
B
D
A
C
B
D
© Glencoe/McGraw-Hill A21 Glencoe Geometry
Chapter 3 Assessment Answer KeyForm 2A (continued) Form 2BPage 164 Page 165 Page 166
An
swer
s
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
C
A
B
D
A
A
C
B
D
C
Sample answer:y � 285 � 15x;
19 h
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
D
A
A
B
C
B
D
A
C
C
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
D
B
B
A
C
A
D
B
C
A
a ⊥ c
© Glencoe/McGraw-Hill A22 Glencoe Geometry
Chapter 3 Assessment Answer KeyForm 2CPage 167 Page 168
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
SV���
Sample answer:
V�Z�
alternate exterior
alternate interior
corresponding
94
x � 13, y � 39
�35
�
�6
� �53
�
neither
parallel
parallel
C � 1.18p � 12;$71
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
B:
y � �9x � 3
y � 3x � 8
y � �13
�x � 1
y � �x � 2
AC��� || DF���; alt. int. �
AD��� || BE���; corr. �
AD��� || BE���; cons. int. �
18
�98� or 7�2��45� or 3�5��20� or 2�5�
Sample answer:Given: �3 and �4are supplementary
Prove: � || m
1
43
ms
�
B
x
y
O
�
© Glencoe/McGraw-Hill A23 Glencoe Geometry
Chapter 3 Assessment Answer KeyForm 2DPage 169 Page 170
An
swer
s
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
CD���
Sample answer:A�E�
corresponding
alternate interior
alternate exterior
91
x � 17; y � 26
��25
�
�17
�
��13
�
perpendicular
parallel
perpendicular
T � 0.04c � 25;$85
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
B:
y � 7x � 8
y � �4x � 8
y � �25
�x � 2
y � �2x � 9
RS��� || TU���; corr. �
PT��� || QU���; alt. int. �
PT��� || QU���; cons. int. �
12
�10��26�
�32� or 4�2�
Sample answer:Given: �2 � �3
Prove: � || m
1
32
mt
�
x
y
O
�
Q
© Glencoe/McGraw-Hill A24 Glencoe Geometry
Chapter 3 Assessment Answer KeyForm 3Page 171 Page 172
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Sample answer:
Sample answer:Plane ABC
intersects planeBCE at BC���.
Sample answer: AD���
is skew to EF���.
corresponding
consecutive interior
alternate exterior
40
x � 18, y � 7.5,and z � �7
�53
�
��38
�
perpendicular
parallel
B
C
F
D
A
E
13.
14.
15.
16.
17.
18.
19.
20.
B:
y � 10 � �85
�(x � 6)
or y � �85
�x � �25
�
y � ��43
�x � �331�
C � 0.28(m � 100)� 150; $193.40
NU��� || PV���; alt. int. �
MR��� || NS���; cons. int. �
24.2
�17�
�20� or 2�5�
Sample answer: No, if
line p and line q lie indifferent planes, thelines could be skew
and not parallel.
sp
q
x
y
OP
m
Chapter 3 Assessment Answer KeyPage 173, Open-Ended Assessment
Scoring Rubric
© Glencoe/McGraw-Hill A25 Glencoe Geometry
Score General Description Specific Criteria
• Shows thorough understanding of using the relationshipsbetween lines and their properties, proving lines parallel,identifying and determining angle relationships, findingslope, and graphing parallel and perpendicular lines.
• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Graphs and figures are accurate and appropriate.• Goes beyond requirements of some or all problems.
• Shows an understanding of using the relationshipsbetween lines and their properties, proving lines parallel,identifying and determining angle relationships, findingslope, and graphing parallel and perpendicular lines.
• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Graphs and figures are mostly accurate and appropriate.• Satisfies all requirements of problems.
• Shows a partial understanding of using the relationshipsbetween lines and their properties, proving lines parallel,identifying and determining angle relationships, findingslope, and graphing parallel and perpendicular lines.
• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Graphs and figures are mostly accurate.• Satisfies the requirements of most of the problems.
• Final computation is correct.• No written explanations or work is shown to substantiate
the final computation.• Graphs and figures may be accurate but lack detail or
explanation.• Satisfies minimal requirements of some of the problems.
• Shows little or no understanding of using the relationshipsbetween lines and their properties, proving lines parallel,identifying and determining angle relationships, findingslope, and graphing parallel and perpendicular lines.
• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Graphs and figures are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer may be given.
0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given
1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation
2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem
3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation
4 SuperiorA correct solution that is supported by well-developed, accurateexplanations
An
swer
s
© Glencoe/McGraw-Hill A26 Glencoe Geometry
Chapter 3 Assessment Answer KeyPage 173, Open-Ended Assessment
Sample Answers
1a. If you know the measure of one or moreangles formed by the intersections of thestreets, you can determine congruence ofangles. If certain angles are congruent,the streets are parallel. For instance,lines a, b, and c are cut by transversal d.If you know the measure of �1 and it iscongruent to �2, then the streetsrepresented by a and b are parallel bythe Alternate Exterior Angles Theorem.If �2 is congruent to �4, b and c areparallel by the Alternate Interior AnglesTheorem.
b. m�4 � 68. Given lines d and e areparallel, �5 and �4 are consecutiveinterior angles. Since consecutiveinterior angles are supplementary,180 � m�5 will give the measure of 4:180 � 112 � 68.
c. x � 27; �1 and �4 are correspondingangles. If lines a and c are parallel, then�1 � �4. Therefore, m�1 � m�4.Substitute the measures for the anglesand solve for x and then verify that theangles are congruent. Since m�1 � 3x � 7 or 74 and m�4 � 2x � 20 or 74; �1 � �4 andlines a and c are parallel.
d. Student should draw a line through ●●Bperpendicular to line b. The perpendicularline segment should include a right anglesymbol. To find the shortest distancefrom the library to his street, Todd couldmeasure this perpendicular segmentsince a perpendicular line segment fromany point to a line is the shortestdistance between the point and the line.
2a. The student first draws the line, thenwrites the equation y � x � 2.
m � �yx
2
2
�
�
yx
1
1� Use the slope formula to find the slope
of the line.
� �42
��
13� Substitute coordinates into the formula.
� �55� or 1
Then use one of the pair of coordinates towrite the equation in point-slope form.
y � y1 � m(x � x1) Point-slope form
y � 1 � 1(x �3) Substitute (�3, �1).
y � x � 3 � 1 Simplify and subtract 1 from
each side.
y � x � 2
b. The slope of a line parallel to y � x � 2is 1, since parallel lines have the sameslope.
c. �2�; the student should draw a lineperpendicular to the two lines, and thenuse the points of intersections and theDistance Formula to find the distance.
In addition to the scoring rubric found on page A25, the following sample answers may be used as guidance in evaluating open-ended assessment items.
© Glencoe/McGraw-Hill A27 Glencoe Geometry
Chapter 3 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 174 Page 175 Page 176
An
swer
s
Quiz 2Page 175
Quiz 4Page 176
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
false; alternateinterior angles
false; Corr. �Postulate
true
false; two
true
true
false; parallel
false;perpendicular
true
false; slope
Sample answer:Rate of change
describes how aquantity is changing
over time.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Sample answer:plane ABM
BC���
corresponding
alternate interior
alternate exterior
consecutive interior
86
94
x � 12, y � 31
C
1.
2.
3.
4.
5.
�1
5
��32
�
�15
�
perpendicular
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
y � 8 � ��13
�(x � 3)
y � �53
�x � 2
y � �4x � 3
C � 5b � 8
g || h; corr. �
p || q; alt. int. �
g || h; alt. ext. �
p || q; cons. int. �
116
26
1.
2.
3.
4.
�32� or 4�2�
12
�2�
x
y
O
�H
BA
CD
© Glencoe/McGraw-Hill A28 Glencoe Geometry
Chapter 3 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 177 Page 178
Part I
Part II
6.
7.
8.
9.
10.
x � 28, y � 19
�12
�
��14
�
parallel
perpendicular
1.
2.
3.
4.
5.
D
C
C
A
B
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
7; 84
(�7, �1)
acute
5; 20
true
H: m�1 � m�2 �180; C: �1 and �2are supplementary
Law of Detachment
always
parallel
y � 8 � 4(x � 2)
�41� or � 6.4
© Glencoe/McGraw-Hill A29 Glencoe Geometry
An
swer
s
Chapter 3 Assessment Answer KeyStandardized Test Practice
Page 179 Page 180
1.
2.
3.
4.
5.
6.
7.
8.
9.
10. E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
11. 12.
13.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
14.
15.
16.
17.
18.
10
5 or �9
2
2
20
2 3 8 5
3
© Glencoe/McGraw-Hill A30 Geometry
Chapter 3 Assessment Answer KeyUnit 1 Test/Review
Page 181 Page 182
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
2�12
� in.; �14
� in.
�68� or about8.25; (9, �8)
m�PQR � 120,obtuse;
m�RQS � 40, acute;m�SQT � 20, acute
53 and 37
pentagon,convex, irregular
54 ft long by 18 ft wide
U
hypothesis: in aplane, lines � andm are equidistant
from line p;conclusion: � || m
Law of Syllogism
sometimes
p q �p �p � q
T T F FT F F FF T T TF F T F
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Given
Angles comp. to� � are �.
Def. of � �
Def. of comp. �
Substitution
p; alternateexterior angles
m�2 � 52, m�4 �128, m�10 � 52,
m�12 � 128
perpendicular
y � �2x � 5
12
�40� or about6.32 units
x
y
O