SECTION Ready to Go On? Skills Intervention 6A 6-1 .... 6...Finding Interior Angle Measures and Sums...

Find these vocabulary words in Lesson 6-1 and the Multilingual Glossary.

Identifying PolygonsTell whether each figure is a polygon. If it is a polygon, name it by the number of its sides.

Is the figure closed ? Is it a plane figure?

Is the figure formed by three or more segments?

Do the segments intersect only at their endpoints?

Are any of the segments with common endpoints collinear?

Is the figure a polygon? How many sides does it have?

Name the polygon by the number of sides it has.

Finding Interior Angle Measures and Sums in Polygons

A. Find the sum of the interior angle measures of a convex 15-gon. The sum of the interior angle measures of a convex polygon with

n sides is (n � )180�. What does the n represent?

Substitute 15 for n in the formula. ( � 2)180�

Simplify the expression.

B. Find the measure of each interior angle of a regular 15-gon.

What is a regular polygon?

From Part A, the sum of the interior angle measures of a 15-gon is . How do you find the measure of each interior angle of a regular polygon?

To find the measure of each interior angle of a regular 15-gon,

divide 2340� by . 2340� ______ �

Ready to Go On? Skills Intervention6-1 Properties and Attributes of Polygons

Vocabulary

side of a polygon vertex of a polygon diagonal

regular polygon concave convex

Copyright © by Holt, Rinehart and Winston. 76 Holt GeometryAll rights reserved.

Name Date Class

SECTION

6A

The Polygon Exterior Angle Sum Theorem states that the sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360�.

A decorative garden in the shape of a pentagon is surrounded by five railings. Find the measure of each exterior angle of the garden.

Understand the Problem

1. What is a pentagon?

2. What is an exterior angle?

Make a Plan

3. The Polygon Exterior Angle Sum Theorem states that the sum of the exterior angle

measures, one angle at each vertex, of a convex polygon is .

4. How will you find the measure of each exterior angle?

Solve

5. Substitute the given angle measures into the Polygon Exterior Angle Sum Theorem.

� 17x � � 12x � � 360�

6. Combine like terms and solve for x.

7. Substitute the value of x into each given angle measure.

Look Back

8. Find the sum of the angle measures you found in Exercise 7.

9. What is the sum of exterior angle measures of a convex polygon?

10. Is your answer in Exercise 8 the same as your answer from Exercise 9?

6AReady to Go On? Problem Solving Intervention6-1 Properties and Attributes of Polygons

Name Date Class


SECTION

A

B

C D

E

12x°

14x° 8x°

17x°

9x°

Find this vocabulary word in Lesson 6-2 and the Multilingual Glossary.

Using Properties of Parallelograms to Find Measures�PQRS is a parallelogram. PT � 53, PS � 76, and m�QRS � 75�.Find each measure.

A. RT

In a parallelogram, the diagonals

each other, so _

PT � and PT � .

Since PT � 53, RT � .

B. QR

In a parallelogram, opposite sides are , so _

QR � ,

and QR � . Since PS � 76, QR � .

C. m�RQP

In a parallelogram, consecutive angles are .

m�RQP � m�QRS �

Substitute 75� for m�QRS. m�RQP � �

Solve to find m�RQP. m�RQP �

D. m�RSP

In a parallelogram, angles are congruent. �RSP �

m�RSP � . What is m�RQP ?

Substitute to find m�RSP. m�RSP �

E. m�QPS

m�QPS � . Substitute to find m�QPS. m�QPS �

F. RS

In a parallelogram, opposite sides are congruent so _

RS � , and RS � .

Substitute the given values for RS and QP. � 7x � 41

Subtract 4x from both sides. � 3x � 41

Add 41 to both sides. 42 �

Divide both sides by 3. x �

Substitute your solution into 7x � 41 and simplify to find RS. 7( ) � 41 �

Ready to Go On? Skills Intervention6-2 Properties of Parallelograms


Name Date Class

SECTION

6A

Vocabulary

parallelogram

P

T

S

Q R

4x + 1 7x – 41

If a quadrilateral is a parallelogram, then its opposite sides are congruent.

The parking spots in a parking lot are in the shape of a parallelogram. If m�DAB � 65� and AE � 13.3 ft, find m�CDA, m�CBA, EC, DC, and DA.


1. Why is it important that the problem states that the parking spots

are in the shape of parallelograms?

Make a Plan

2. What do you know about consecutive angles of a parallelogram?

3. What do you know about opposite angles of a parallelogram?

4. What do you know about opposite sides of a parallelogram?

5. What do you know about the diagonals of a parallelogram?

Solve

6. Find m�CDA. Explain how you determined your answer.

7. Find m�CBA. Explain how you determined your answer.

8. Find EC. Explain how you determined your answer.

9. Find DC and DA. Explain how you determined your answer.

Look Back

10. Do your answers to Exercises 6–9 satisfy all of the properties of parallelograms

from Exercises 2–5?

6AReady to Go On? Problem Solving Intervention6-2 Properties of Parallelograms

Name Date Class


SECTION

C

D

A

BE

20 ft

11 ft

Ready to Go On? Skills Intervention6-3 Conditions for Parallelograms

Verifying Figures are ParallelogramsShow that �JKLM is a parallelogram for x � 9 and y � 2.5.

For �JKLM to be a parallelogram, both pairs of

have to be when x � 9 and y � 2.5.

Step 1 Find JM and KL.

JM � KL �

Substitute x � 9 into each expression and simplify.

JM � 4( ) � 6 KL � 6( ) � 12

JM � KL �

Since JM � KL, _

JM _

KL .

Step 2 Find JK and ML.

JK � ML �

Substitute y � 2.5 into each expression and simplify.

JK � 10( ) � 3 ML � 14( ) � 7

JK � ML �

Since JK � ML, _

JK _

ML .

Step 3 Both pairs of sides of the quadrilateral are , so

the quadrilateral is a .

Applying Conditions for ParallelogramsDetermine if the quadrilateral must be a parallelogram. Justify your answer.

For a quadrilateral to be a parallelogram you must be able to prove that at least one

pair of sides are parallel and .

The arrows on the opposite sides indicate that the sides are

.

The tic marks on the opposite sides indicate that the sides are

.

Since one pair of sides of a quadrilateral are and

, the quadrilateral is a .


Name Date Class

SECTION

6A

M L

J K10y + 3

14y – 7

6x – 124x + 6

Name Date Class


6-1 Properties and Attributes of PolygonsTell whether each figure is a polygon. If it is a polygon, name it by the number of its sides.

1. 2.

3. 4.

5. Find the sum of the interior angle measures of a convex 21-gon.

6. The surface of a stop sign is in the shape of a regular octagon. Find the

measure of each interior angle of the stop sign.

7. A decorative pool in the shape of a pentagon is bordered by five rows of bushes, as shown. Find the measure of each exterior angle of the pool.

8. Find the measure of each exterior angle of a heptagon.

6-2 Properties of ParallelogramsTiles used to cover a floor are in the shape of a parallelogram. In �PQRS, PS � 3 in., TQ � 3.5 in., and m�RSP � 58�. Find each measure.

9. SQ 10. TS 11. QR

12. m�QPS 13. m�QRS 14. m�RQP

15. Three vertices of parallelogram WXYZ are W(�1, 3), X(6, 4) and Y(4, �1).

Find the coordinates of vertex Z.

Ready to Go On? Quiz6A

SECTION

A

B

CD

E

15x°

42x°

13x°

25x°

25x°

S

T

R

P Q

2 4 6–2

2

–2 x

y


Name Date Class

Ready to Go On? Quiz continued

6A SECTION

DEFG is a parallelogram. Find each measure.

16. FG

17. DE

18. m�F

19. m�E

6-3 Conditions for Parallelograms

20. Show that HIJK is a parallelogram for x � 6.5 and y � 7.

21. Show that ABCD is a parallelogram for x � 11 and y � 8.

Determine if each quadrilateral is a parallelogram. Justify your answer.

22. 23. 24.

25. Show that a quadrilateral with vertices J(�2, 1), K(3, 3), L(�1, �4), and M(�6, �6) is a parallelogram.

D E

G F3y + 11

6y – 1

(3x + 15)�

(7x – 5)�

H K

I J7y + 10

9y – 4

8x – 9 6x + 4

A B

D C

(15y + 2)�

(9x – 41)�

(5x + 3)�

2 4

–4

–2

2

–2–4–6 x

y

Name Date Class


Ready to Go On? Enrichment6A

SECTION

Polygons and ParallelogramsAnswer each question.

1. The measure of an exterior angle of a regular polygon is x �, and the measure of an interior angle is (10x � 15)�. Name the polygon.

2. Find the measure of each angle in the polygon at right if:

m�A � ( x 2 � 7x � 1)�

m�B � (13x � 21)�

m�C � (11x � 4)�

m�D � ( x 2 � 5x � 1)�

m�E � (2 x 2 � x � 1)�

m�F � (17x � 12)�

m�G � ( x 2 � 3x � 4)�

3. The coordinates of three vertices of a parallelogram are (1, 5), (4, 3) and (2, �2). Find the coordinates of two other possible locations of the fourth vertex.

4. ABCD is a parallelogram. m�A � (7y � x)�, m�B � (2x � 5)�, and m�D � (3y � 12)�. Find the measure of each angle.

5. The diagonals of a parallelogram intersect at (1, 1). Two vertices are located at (�6, 4) and (�3, �1). Find the coordinates of the other two vertices.

AB

C

D EF

G

2 4 6

–4

–6

–2

2

4

6

–2–4–6 x

yA B

CD


Using Properties of Rectangles to Find MeasuresPQRS is a rectangle. PQ � 64 ft and PR � 70 ft. Find each measure.

A. SR

_

PR � Rectangle opposite sides �

PQ � � 64 Definition of � segments

So, SR � .

B. TQ

_

PR � Rectangle � diagonals

PR � QS � Definition of � segments

TQ � 1 __ 2 � � Parallelogram diagonals bisect each other

TQ � 1 __ 2 � � Substitute and simplify.

� ft

Using Properties of Rhombuses to Find MeasuresCDEF is a rhombus. Find DE.

CD � FC Definition of a rhombus

10x � 9 � Substitute the given values.

10x � 9 � 22 � 15x � 22 � 22 Add 22 to both sides.

10x � � 15x

� 10x � � 15x � Subtract 10x from both sides.

� 5x Divide both sides by 5.

� x

DE � Definition of a rhombus

DE � Substitute 15x � 22 for FC.

DE � 15 ( ) � 22 Substitute 6.2 for x.

DE � Simplify.


Name Date Class

Ready to Go On? Skills Intervention6-4 Properties of Special Parallelograms6B

SECTION

Vocabulary

rectangle rhombus square

S R

T

P Q

10x + 9

15x – 22

E F

GD C

Name Date Class


If a parallelogram is a rectangle, then its diagonals are congruent.

A rectangular door is inset with glass that has decorative strips along the diagonals. In rectangle FGHI, FG � 24 in. and FH � 65 in. Find each length.

A. HI

B. GI

C. JH


1. Why is it important to know the shape of the door?

2. What measurements of the rectangle are important to finding the lengths?

Make a Plan

3. How are opposite sides of a rectangle related?

4. How are the diagonals of a rectangle related?

Solve

5. If FG � 24 in., what do you know about HI ?

6. Since FH � 65 in., what do you know about GI?

7. Since FH � 65 in., what do you know about JH ?

So, JH �

Look Back

Use the properties of rectangles to see if your answers are logical.

8. Are FG and HI opposite sides of a rectangle? Do they have the same length?

9. Are FH and GI diagonals of a rectangle? Do they have equal length?

10. Do diagonals of a rectangle bisect each other? So, is 2 � JH � FH?

Ready to Go On? Problem Solving Intervention6-4 Properties of Special Parallelograms6B

SECTION

GF

H

OJ

I

Applying Conditions for Special ParallelogramsDetermine if the conclusion is valid. If not, tell what additional information is needed to make it valid.

Given: _

PS � _

QR , �SPR � �QRT. _

PR � _

QS

Conclusion: PQRS is a rhombus.

Step 1 Determine if PQRS is a parallelogram.

�SPR � �QRT Given

_

PS � _

QR Converse of Theorem

_

PS � _

QR

PQRS is a parallelogram. Quad with one pair of sides that are and parallelogram

Step 2 Determine if PQRS is a rhombus.

_

PR � _

QS

PQRS is a rhombus. Parallelogram with diagonals rhombus

Step 3 Since PQRS is a with diagonals, it is a

. Is the conclusion valid?

Identifying Special Parallelograms in the Coordinate PlaneUse the diagonals to determine whether the parallelogram with the given vertices is a rectangle, rhombus, or square. Give all names that apply.

J(1, 4), K(6, 1), L(3, �4), M(�2, �1)

Step 1 Graph JKLM on the grid at right.

Step 2 Use the Distance Formula to find JL and MK to determine if JKLM is a rectangle.

JL � ��

� 1 � � 2 � � 4 � � 2 � ��

� 2 ��

KM � ��

� � 6 � 2 � � � 1 � 2 � ��

� 2 ��

Since JL KM, JKLM is a .

Step 3 Find the slopes of _

JL and _

KM to tell if JKLM is a rhombus.

slope of _

JL � �4 � ________ 3 �

� ____ � slope of _

KM � � 1 ________ � 6

� ____ � ___

Since � � � 1 __ 4 � � �1, _

JL _

KM . JKLM is a .

Since JKLM is a and a , JKLM is also a .


Name Date Class

Ready to Go On? Skills Intervention6-5 Conditions for Special Parallelograms6B

SECTION

P

R

TQ S

2 4 6

–4

–6

–2

2

4

6

–2–4–6 x

y

Name Date Class



Using Properties of KitesIn kite WXYZ, m�VWX � 85� and m�VWZ � 48�. Find each measure.

A. m�WXV

m�WVX � Kite Diagonals are

m�VWX � m�VXW � Acute angles of a right triangle are .

� m�VXW � 90 Substitute 85 for m�VWX.

m�VXW � Subtract 85 from both sides.

B. m�XYZ

�ZWX � Kite one pair of opposite angles.

m�ZWX � Definition of angles

m�ZWX � m�VWZ � m�VWX Angle Postulate

m�ZWX � � Substitute 48° for m�VWZ and 85° for m�VWX.

m�ZWX � Simplify.

m�XYZ � Substitute m�ZYX for m�ZWX.

Using Properties of Isosceles TrapezoidsIf CA � 53, DE � 20, find EB.

_

DB � Isosceles trapezoid diagonals congruent

DB � Definition of congruent

DB � Substitute for CA.

DE � EB � DB Addition Postulate

� EB � Substitute 20 for DE and 53 for DB.

EB � Subtract 20 from both sides.

Ready to Go On? Skills Intervention6-6 Properties of Kites and Trapezoids6B

SECTION

Vocabulary

kite trapezoid base of a trapezoid leg of a trapezoid

base angle of a trapezoid isosceles trapezoid midsegment of a trapezoid

W

Y

V

X

Z

D C

BAE


Name Date Class

Ready to Go On? Problem Solving Intervention6-6 Properties of Kites and Trapezoids6B

SECTION

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs.

The front of a decorative end table is in the shape of a trapezoid. The bases are 37 cm and 54 cm long. The bottom of the top drawer extends from the midpoint of each leg of the trapezoid. How long is the bottom of the top drawer?


1. The of a trapezoid is the segment whose endpoints are the

midpoints of the .

2. The midsegment of a trapezoid is to each base, and its length is

the sum of the lengths of the .

Make a Plan

3. Apply the Midsegment Theorem, using and for the lengths of the bases.

Solve

4. Substitute the lengths of bases into the Midsegment Theorem and simplify.

� _________ 2

____ 2

5. The length of the bottom of the drawer is .

Look Back

6. Work backwards from your answer to check your solution. Multiply your answer in Exercise 6 by 2.

7. What is the sum of the bases of the trapezoid?

8. Do your answers in Exercises 7 and 8 match?

Name Date Class


6-4 Properties of Special ParallelogramsThe flag of Florida is a rectangle with stripes along the diagonals. In rectangle ABCD, AD � 45 in. and BD � 52.5 in. Find each length.

1. ED 2. AC

3. BC 4. EC

JKLM is a rhombus. Find each measure.

5. MJ

6. Find m�MJN and m�LMJ if m�MNJ � (6d � 12)� and m�NKJ � (4d �1)�.

7. Given: ADCE is a rhombus with diagonal _

ED . _

CB � _

AB . Prove: �BCD � �BAD.

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

6-5 Conditions for Special ParallelogramsDetermine if the conclusion is valid. If not, tell what additional information is needed to make it valid.

8. Given: _

WY � _

XZ , _

WZ � _

WX Conclusion: WXYZ is a rhombus.

9. Given: _

WX � _

ZY , _

WZ � _

XY , _

WZ � _

ZY Conclusion: WXYZ is a rectangle.

Ready to Go On? Quiz6B

SECTION

A

E

D

B C

8x + 5

12x – 8M L

N

J K

E

C

A B

D

W X

Z Y


Name Date Class


6BSECTION

2 4 6

–4

–6

–2

2

4

6

–2–4–6 x

yUse the diagonals to determine whether a parallelogram with the given vertices is a rectangle, a rhombus, or a square. Give all the names that apply.

10. H(3, 5), I(�1, 2), J(�3, �4), K(1, �1)

11. P(2, 4), Q(4, �2), R(�2, �4), S(�4, 2)

12. Given: �MON is equilateral. M is the midpoint of _

LN . LMOP is a parallelogram. Prove: LMOP is a rhombus.

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6-6 Properties of Kites and TrapezoidsIn kite CDEF, m�CDF � 39�, and m�EFC � 25�. Find each measure.

13. m�CFG 14. m�GEF

15. m�DCG 16. m�DEF

17. Find m�Q. 18. AC � 91.7 and BE � 33.9. Find ED.

54°

Q

P S

R

E

A B

CD

19. The face of a stone wall is in the shape of a trapezoid. The bases of the wall are 132 in. and 64 in. A steel bar is attached between the midpoint of each leg of the trapezoid. How long is

the bar?

P

L M N

O

E

D

C

G

F

Name Date Class


Other Special Quadrilaterals

1. Given: ABCD is a rectangle. E, F, G, and H are midpoints of their respective sides. Write a paragraph proof to show that EFGH is a rhombus.

2. The quadrilateral at right is a kite, not drawn to scale. Which two sides are congruent,

_ RS and

_ RQ or

_ RS

and _

ST ? Why?

Ready to Go On? Enrichment6B

SECTION

A

E

D

G

B F C

H

Q T

4x + 315 – 3x

2x + 35R S



5-6 Inequalities in Two Triangles

7. Compare AB and ST. 8. Compare m�XWY 9. Find the range of values and m�ZWY. for x.

C B RS

TA

120° 100°33

33

29

29

W

Z

X

Y

88

8844

40

64

74 5x – 11

55°

40°

ST � AB m�XWY � m�ZWY x � 17

5-7 The Pythagorean Theorem

10. Find the value of x. Give the answer in simplest radical form.

��

442

11. Tell if the measures 8, 9, and 15 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.

The sides form an acute triangle because 8 2 � 9 2 � 15 2 .

12. A park developer want to put a bike trail from one corner of a rectangular park to the opposite corner. What will be the length of the trail? Round to the nearest yard.

832 yd

5-8 Applying Special Right Triangles

13. A decorative platter is an equilateral triangle with side lengths of 14 inches. What is the height of the platter? Round to the nearest inch.

12 in.

Find the values of the variables. Give your answers in simplest radical form.

14. 15. 16.

11

45° x

x60°

y12 ��

3

4045°

x

x

x � 11 ��

2 x � 12, y � 24 x � 20 ��

2


5BSECTION

9

x 19

14 in.

60° 60°

450 yd

700 yd

Bike tra

il

057-075_CH05_RTGO_GEO_12738.indd 74 5/25/06 4:30:31 PM

Relationships in TrianglesFor Exercises 1–2, tell whether a triangle can have vertices with the given coordinates. Explain.

1. �PQR has vertices P(�3, 11), Q(1, 3) and R(5, �5).

No, because PQ � QR � 4 ��

5 , and PR � 8 ��

5 . Does not satisfy the � Ineq.

2. �JKL has vertices J(�5, 1), K(4, 2) and L(11, �1).

Yes, because JK � ��

82 , KL � ��

58 , and JL � 2 ��

65 .

Satisfies � Ineq.

Answer each question.

3. A right triangle has legs with lengths x and 3(x � 1), and hypotenuse 4x – 3. Find x and the lengths of each side.

x � 7; The lengths of the sides are 7, 24, and 25.

4. The figure at the right is drawn to scale. Compare BC and AD. Which segment is longer? Explain your answer.

Since �BCA is acute and �ACD is its

supplement, �ACD must be obtuse.

Therefore AD � BC.

5. In the figure at the right, m�RPS � 53�. _

PQ � _

RS , and PQ � PR. Compare QS and RS. Explain your answer.

RS � QS; By Alt Int �’s, m�QPS � 28�, so

m�RPS � m�QPS and PS � PS; By the Hinge

Theorem, RS � QS.

6. �MNP is an equilateral triangle. _

RM � _

RP . MQ � 3 ��

2 . Find the following:

NQ ��

6

MN 2 ��

6

RP 2 ��

3


SECTION

B

C

D

A

53°

28°R

S

QP

RM

P

QN




Identifying PolygonsTell whether each figure is a polygon. If it is a polygon, name it by the number of its sides.

Is the figure closed ? Yes Is it a plane figure? Yes

Is the figure formed by three or more segments? Yes

Do the segments intersect only at their endpoints? Yes

Are any of the segments with common endpoints collinear? No

Is the figure a polygon? Yes How many sides does it have? 12

Name the polygon by the number of sides it has. Dodecagon

Finding Interior Angle Measures and Sums in Polygons

A. Find the sum of the interior angle measures of a convex 15-gon. The sum of the interior angle measures of a convex polygon with

n sides is (n � 2 )180�. What does the n represent?

The number of sides in the polygon

Substitute 15 for n in the formula. ( 15 � 2)180�

Simplify the expression. 2340�

B. Find the measure of each interior angle of a regular 15-gon.

What is a regular polygon? A polygon that is both equiangular and equilateral.

From Part A, the sum of the interior angle measures of a 15-gon is 2340� . How do you find the measure of each interior angle of a regular polygon?

Divide the sum of the interior angle measures by the number of sides. To find the measure of each interior angle of a regular 15-gon,

divide 2340� by 15 . 2340� ______

15 � 156�

Ready to Go On? Skills Intervention6-1 Properties and Attributes of Polygons

Vocabulary

side of a polygon vertex of a polygon diagonal

regular polygon concave convex


SECTION

6A


The Polygon Exterior Angle Sum Theorem states that the sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360�.

A decorative garden in the shape of a pentagon is surrounded by five railings. Find the measure of each exterior angle of the garden.


1. What is a pentagon? A polygon with five sides

2. What is an exterior angle? An angle formed by one side of a polygon

and the extension of a consecutive side.

Make a Plan

3. The Polygon Exterior Angle Sum Theorem states that the sum of the exterior angle

measures, one angle at each vertex, of a convex polygon is 360� .

4. How will you find the measure of each exterior angle? Solve for x, and substitute

the value of x into each angle measure.

Solve

5. Substitute the given angle measures into the Polygon Exterior Angle Sum Theorem.

8x � 17x � 9x � 12x � 14x � 360�

6. Combine like terms and solve for x. 60x � 360�, x � 6

7. Substitute the value of x into each given angle measure. (Exterior angles named by the

vertex angle) m�A � 48�, m�B � 84�, m�C � 72�, m�D � 54�, m�E � 102�

Look Back

8. Find the sum of the angle measures you found in Exercise 7. 360�

9. What is the sum of exterior angle measures of a convex polygon? 360�

10. Is your answer in Exercise 8 the same as your answer from Exercise 9? Yes

6AReady to Go On? Problem Solving Intervention6-1 Properties and Attributes of Polygons


SECTION

A

B

C D

E

12x°

14x ° 8x°

17x°

9x °

076-091_CH06_RTGO_GEO_12738.indd 77 12/15/05 9:43:36 AM


Find this vocabulary word in Lesson 6-2 and the Multilingual Glossary.

Using Properties of Parallelograms to Find Measures�PQRS is a parallelogram. PT � 53, PS � 76, and m�QRS � 75�.Find each measure.

A. RT

In a parallelogram, the diagonals bisect

each other, so _

PT � _

RT and PT � RT .

Since PT � 53, RT � 53 .

B. QR

In a parallelogram, opposite sides are congruent , so _

QR � PS ,

and QR � PS . Since PS � 76, QR � 76 .

C. m�RQP

In a parallelogram, consecutive angles are supplementary .

m�RQP � m�QRS � 180�

Substitute 75� for m�QRS. m�RQP � 75� � 180�

Solve to find m�RQP. m�RQP � 105�

D. m�RSP

In a parallelogram, opposite angles are congruent. �RSP � �RQP

m�RSP � m�RQP . What is m�RQP ? 105�

Substitute to find m�RSP. m�RSP � 105�

E. m�QPS

m�QPS � m�QRS . Substitute to find m�QPS. m�QPS � 75�

F. RS

In a parallelogram, opposite sides are congruent so _

RS � QP , and RS � QP .

Substitute the given values for RS and QP. 4x � 1 � 7x � 41

Subtract 4x from both sides. 1 � 3x � 41

Add 41 to both sides. 42 � 3x

Divide both sides by 3. x � 14

Substitute your solution into 7x � 41 and simplify to find RS. 7( 14 ) � 41 � 57

Ready to Go On? Skills Intervention6-2 Properties of Parallelograms


SECTION

6A

Vocabulary

parallelogram

P

T

S

Q R

4x + 1 7x – 41


If a quadrilateral is a parallelogram, then its opposite sides are congruent.

The parking spots in a parking lot are in the shape of a parallelogram. If m�DAB � 65� and AE � 13.3 ft, find m�CDA, m�CBA, EC, DC, and DA.


1. Why is it important that the problem states that the parking spots

are in the shape of parallelograms? So you can find the

missing lengths using the properties of parallelograms.

Make a Plan

2. What do you know about consecutive angles of a parallelogram?

Consecutive angles of a parallelogram are supplementary.

3. What do you know about opposite angles of a parallelogram?

Opposite angles of a parallelogram are congruent.

4. What do you know about opposite sides of a parallelogram?

Opposite sides of a parallelogram are congruent.

5. What do you know about the diagonals of a parallelogram?

The diagrams of a parallelogram bisect each other.

Solve

6. Find m�CDA. Explain how you determined your answer. m�CDA � 115� because

�CDA and �DAB are supplementary.

7. Find m�CBA. Explain how you determined your answer. m�CBA � 115� because

�CBA and �CDA are opposite angles of the parallelogram.

8. Find EC. Explain how you determined your answer. EC � 13.3; Diagonals of a

parallelogram bisect each other so C is the midpoint of AC.

9. Find DC and DA. Explain how you determined your answer. DC � 20 ft and DA � 11 ft;

Opposite sides of a parallelogram are congruent so they have equal length.

Look Back

10. Do your answers to Exercises 6–9 satisfy all of the properties of parallelograms

from Exercises 2–5? Yes

6AReady to Go On? Problem Solving Intervention6-2 Properties of Parallelograms


SECTION

C

D

A

BE

20 ft

11 ft


Ready to Go On? Skills Intervention6-3 Conditions for Parallelograms

Verifying Figures are ParallelogramsShow that �JKLM is a parallelogram for x � 9 and y � 2.5.

For �JKLM to be a parallelogram, both pairs of opposite sides

have to be congruent when x � 9 and y � 2.5.

Step 1 Find JM and KL.

JM � 4x � 6 KL � 6x � 12

Substitute x � 9 into each expression and simplify.

JM � 4( 9 ) � 6 KL � 6( 9 ) � 12

JM � 42 KL � 42

Since JM � KL, _

JM � _

KL .

Step 2 Find JK and ML.

JK � 10y � 3 ML � 14y � 7

Substitute y � 2.5 into each expression and simplify.

JK � 10( 2.5 ) � 3 ML � 14( 2.5 ) � 7

JK � 28 ML � 28

Since JK � ML, _

JK � _

ML .

Step 3 Both pairs of opposite sides of the quadrilateral are congruent , so

the quadrilateral is a parallelogram .

Applying Conditions for ParallelogramsDetermine if the quadrilateral must be a parallelogram. Justify your answer.

For a quadrilateral to be a parallelogram you must be able to prove that at least one

pair of opposite sides are parallel and congruent .

The arrows on the opposite sides indicate that the sides are

parallel .

The tic marks on the opposite sides indicate that the sides are

congruent .

Since one pair of opposite sides of a quadrilateral are parallel and

congruent , the quadrilateral is a parallelogram .


SECTION

6A

M L

J K10y + 3

14y – 7

6x – 124x + 6



6-1 Properties and Attributes of PolygonsTell whether each figure is a polygon. If it is a polygon, name it by the number of its sides.

1. Yes, hexagon 2. Not a polygon

3. Yes, quadrilateral 4. Not a quadrilateral

5. Find the sum of the interior angle measures of a convex 21-gon. 3420�

6. The surface of a stop sign is in the shape of a regular octagon. Find the

measure of each interior angle of the stop sign. 135�

7. A decorative pool in the shape of a pentagon is bordered by five rows of bushes, as shown. Find the measure of each exterior angle of the pool.

x � 3, measures of exterior angles (named

using 1 letter) m�A � 45�, m�B � 75�,

m�C � 75�, m�D � 39�, m�E � 126�

8. Find the measure of each exterior angle of a heptagon. � 360 ____

7 � � � 51.43�

6-2 Properties of ParallelogramsTiles used to cover a floor are in the shape of a parallelogram. In �PQRS, PS � 3 in., TQ � 3.5 in., and m�RSP � 58�. Find each measure.

9. SQ 7 in. 10. TS 3.5 in. 11. QR 3 in.

12. m�QPS 122� 13. m�QRS 122� 14. m�RQP 58�

15. Three vertices of parallelogram WXYZ are W(�1, 3), X(6, 4) and Y(4, �1).

Find the coordinates of vertex Z. (�3, �2)

Ready to Go On? Quiz6A

SECTION

A

B

CD

E

15x °

42x°

13x°

25x°

25x °

S

T

R

P Q

2 4 6–2

2

–2 x

y





6A SECTION

DEFG is a parallelogram. Find each measure.

16. FG 23

17. DE 23

18. m�F 66�

19. m�E 114�

6-3 Conditions for Parallelograms

20. Show that HIJK is a parallelogram for x � 6.5 and y � 7.

8(6.5) � 9 � 6(6.5) � 4 � 43. 7(7) � 10 �

9(7) � 4 � 59. Since both pairs of opposite

sides have equal length, HIJK is a parallelogram.

21. Show that ABCD is a parallelogram for x � 11 and y � 8.

9(11) � 41 � 5(11) � 3 � 58�. 15(8) � 2 � 122�.

Since �CBA is supplementary to both of its

consecutive angles, ABCD is a parallelogram.

Determine if each quadrilateral is a parallelogram. Justify your answer.

22. 23. 24.

Yes, both pairs of No, not enough Yes, one pair of

opposite sides are �. information given opposite sides � and �

25. Show that a quadrilateral with vertices J(�2, 1), K(3, 3), L(�1, �4), and M(�6, �6) is a parallelogram.

The slope of JK and the slope of LM are both 2 _

5 and the slope of

KL and the slope of MJ are both 7 _

4 . Since the quadrilateral has

two pair of opposite parallel sides, it is a parallelogram.

D E

G F3y + 11

6y – 1

(3x + 15)�

(7x – 5)�

H K

I J7y + 10

9y – 4

8x – 9 6x + 4

A B

D C

(15y + 2)�

(9x – 41)�

(5x + 3)�

2 4

–4

–2

2

–2–4–6 x

y



Ready to Go On? Enrichment6A

SECTION

Polygons and ParallelogramsAnswer each question.

1. The measure of an exterior angle of a regular polygon is x �, and the measure of an interior angle is (10x � 15)�. Name the polygon.

24-gon

2. Find the measure of each angle in the polygon at right if:

m�A � ( x 2 � 7x � 1)�

m�B � (13x � 21)�

m�C � (11x � 4)�

m�D � ( x 2 � 5x � 1)�

m�E � (2 x 2 � x � 1)�

m�F � (17x � 12)�

m�G � ( x 2 � 3x � 4)�

m�A � 143�, m�B � 138�, m�C � 95�, m�D � 127�,

m�E � 152�, m�F � 141�, and m�G � 104�

3. The coordinates of three vertices of a parallelogram are (1, 5), (4, 3) and (2, �2). Find the coordinates of two other possible locations of the fourth vertex.

(0, �1) and (3, 10)

4. ABCD is a parallelogram. m�A � (7y � x)�, m�B � (2x � 5)�, and m�D � (3y � 12)�. Find the measure of each angle.

m�D � m�B � 39�, m�A � 141�

5. The diagonals of a parallelogram intersect at (1, 1). Two vertices are located at (�6, 4) and (�3, �1). Find the coordinates of the other two vertices.

(5, 3) and (8, �2)

AB

C

D EF

G

2 4 6

–4

–6

–2

2

4

6

–2–4–6 x

yA B

CD



Using Properties of Rectangles to Find MeasuresPQRS is a rectangle. PQ � 64 ft and PR � 70 ft. Find each measure.

A. SR

_

PR � _

SR Rectangle opposite sides �

PQ � SR � 64 Definition of � segments

So, SR � 64 .

B. TQ

_

PR � _

QS Rectangle � diagonals

PR � QS � 70 Definition of � segments

TQ � 1 __ 2 � SQ � Parallelogram diagonals bisect each other

TQ � 1 __ 2 � 70 � Substitute and simplify.

� 35 ft

Using Properties of Rhombuses to Find MeasuresCDEF is a rhombus. Find DE.

CD � FC Definition of a rhombus

10x � 9 � 15x � 22 Substitute the given values.

10x � 9 � 22 � 15x � 22 � 22 Add 22 to both sides.

10x � 31 � 15x

�10x � 10x � 31 � 15x � 10x Subtract 10x from both sides.

31 � 5x Divide both sides by 5.

6.2 � x

DE � FC Definition of a rhombus

DE � 15x � 22 Substitute 15x � 22 for FC.

DE � 15 ( 6.2 ) � 22 Substitute 6.2 for x.

DE � 71 Simplify.


Ready to Go On? Skills Intervention6-4 Properties of Special Parallelograms6B

SECTION

Vocabulary

rectangle rhombus square

S R

T

P Q

10x + 9

15x – 22

E F

GD C

07 -091_CH0 _ TGO_GEO_12738. 8 12/15/05 9: 3: 0 AM


If a parallelogram is a rectangle, then its diagonals are congruent.

A rectangular door is inset with glass that has decorative strips along the diagonals. In rectangle FGHI, FG � 24 in. and FH � 65 in. Find each length.

A. HI

B. GI

C. JH


1. Why is it important to know the shape of the door? Sample answer:

to use the properties of rectangles to help answer the question.

2. What measurements of the rectangle are important to finding the lengths?

The lengths of the sides and the length of the diagonals.

Make a Plan

3. How are opposite sides of a rectangle related? They are congruent and

have equal measure.

4. How are the diagonals of a rectangle related? They are congruent and

have equal length and they bisect each other.

Solve

5. If FG � 24 in., what do you know about HI ? HI � 24 in.

6. Since FH � 65 in., what do you know about GI? GI � 65 in.

7. Since FH � 65 in., what do you know about JH ? It is one-half of FH.

So, JH � JH � 1 __ 2 FH � 1 __ 2 (65) � 32.5

Look Back

Use the properties of rectangles to see if your answers are logical.

8. Are FG and HI opposite sides of a rectangle? Yes Do they have the same length?

Yes, both equal 24.

9. Are FH and GI diagonals of a rectangle? Yes Do they have equal length? Yes

10. Do diagonals of a rectangle bisect each other? Yes So, is 2 � JH � FH? Yes

Ready to Go On? Problem Solving Intervention6-4 Properties of Special Parallelograms6B

SECTION

GF

H

OJ

I



Applying Conditions for Special ParallelogramsDetermine if the conclusion is valid. If not, tell what additional information is needed to make it valid.

Given: _

PS � _

QR , �SPR � �QRT. _

PR � _

QS

Conclusion: PQRS is a rhombus.

Step 1 Determine if PQRS is a parallelogram.

�SPR � �QRT Given

_

PS � _

QR Converse of Alternate Interior Angles Theorem

_

PS � _

QR Given

PQRS is a parallelogram. Quad with one pair of opp. sides that are � and � parallelogram

Step 2 Determine if PQRS is a rhombus.

_

PR � _

QS Given

PQRS is a rhombus. Parallelogram with perpendicular diagonals rhombus

Step 3 Since PQRS is a parallelogram with perpendicular diagonals, it is a

rhombus . Is the conclusion valid? Yes

Identifying Special Parallelograms in the Coordinate PlaneUse the diagonals to determine whether the parallelogram with the given vertices is a rectangle, rhombus, or square. Give all names that apply.

J(1, 4), K(6, 1), L(3, �4), M(�2, �1)

Step 1 Graph JKLM on the grid at right.

Step 2 Use the Distance Formula to find JL and MK to determine if JKLM is a rectangle.

JL � ��

� 1 � 3 � 2 � � 4 � �4 � 2 � ��

68 � 2 ��

17

KM � ��

� �2 � 6 � 2 � � �1 � 1 � 2 � ��

68 � 2 ��

17

Since JL � KM, JKLM is a rectangle .

Step 3 Find the slopes of _

JL and _

KM to tell if JKLM is a rhombus.

slope of _

JL � �4 � 4 ________

3 � 1 � �8

____ 2

� �4 slope of _

KM � �1 � 1 ________ �2 � 6

� �2 ____

�8 � 1 ___

4

Since � �4 � � 1 __ 4 � � �1, _

JL � _

KM . JKLM is a rhombus .

Since JKLM is a rectangle and a rhombus , JKLM is also a square .


Ready to Go On? Skills Intervention6-5 Conditions for Special Parallelograms6B

SECTION

P

R

TQ S

2 4 6

–4

–6

–2

2

4

6

–2–4–6 x

y

J

K

L

M




Using Properties of KitesIn kite WXYZ, m�VWX � 85� and m�VWZ � 48�. Find each measure.

A. m�WXV

m�WVX � 90� Kite Diagonals are perpendicular

m�VWX � m�VXW � 90� Acute angles of a right triangle are complementary .

85 � m�VXW � 90 Substitute 85 for m�VWX.

m�VXW � 5� Subtract 85 from both sides.

B. m�XYZ

�ZWX � �XYZ Kite one pair of opposite congruent angles.

m�ZWX � m�XYZ Definition of congruent angles

m�ZWX � m�VWZ � m�VWX Angle Addition Postulate

m�ZWX � 48� � 85� Substitute 48° for m�VWZ and 85° for m�VWX.

m�ZWX � 133� Simplify.

m�XYZ � 133� Substitute m�ZYX for m�ZWX.

Using Properties of Isosceles TrapezoidsIf CA � 53, DE � 20, find EB.

_

DB � _

CA Isosceles trapezoid diagonals congruent

DB � CA Definition of congruent segments

DB � 53 Substitute 53 for CA.

DE � EB � DB Segment Addition Postulate

20 � EB � 53 Substitute 20 for DE and 53 for DB.

EB � 33 Subtract 20 from both sides.

Ready to Go On? Skills Intervention6-6 Properties of Kites and Trapezoids6B

SECTION

Vocabulary

kite trapezoid base of a trapezoid leg of a trapezoid

base angle of a trapezoid isosceles trapezoid midsegment of a trapezoid

W

Y

V

X

Z

D C

BAE



Ready to Go On? Problem Solving Intervention6-6 Properties of Kites and Trapezoids6B

SECTION

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs.

The front of a decorative end table is in the shape of a trapezoid. The bases are 37 cm and 54 cm long. The bottom of the top drawer extends from the midpoint of each leg of the trapezoid. How long is the bottom of the top drawer?


1. The midsegment of a trapezoid is the segment whose endpoints are the

midpoints of the legs .

2. The midsegment of a trapezoid is parallel to each base, and its length is

one half the sum of the lengths of the bases .

Make a Plan

3. Apply the Midsegment Theorem, using 37 cm and 54 cm for the lengths of the bases.

Solve

4. Substitute the lengths of bases into the Midsegment Theorem and simplify.

37 � 54

_________ 2

91 ____ 2

45.5

5. The length of the bottom of the drawer is 45.5 cm .

Look Back

6. Work backwards from your answer to check your solution. Multiply your answer in Exercise 6 by 2.

45.5 � 2 � 90

7. What is the sum of the bases of the trapezoid? 90

8. Do your answers in Exercises 7 and 8 match? Yes



6-4 Properties of Special ParallelogramsThe flag of Florida is a rectangle with stripes along the diagonals. In rectangle ABCD, AD � 45 in. and BD � 52.5 in. Find each length.

1. ED 26.25 in. 2. AC 52.5 in.

3. BC 45 in. 4. EC 26.25 in.

JKLM is a rhombus. Find each measure.

5. MJ 31

6. Find m�MJN and m�LMJ if m�MNJ � (6d � 12)� and m�NKJ � (4d �1)�.

m�MJN � 21� and �LMJ � 138�

7. Given: ADCE is a rhombus with diagonal _

ED . _

CB � _

AB . Prove: �BCD � �BAD.

Statements Reasons

1. ADCE is a rhombus. 1. Given

2. _

CB � _

AB 2. Given

3. _

CD � _

AD 3. Def. of a rhombus

4. _

DB � _

DB 4. Refl. Prop. Of �

5. �DAB � �DCB 5. SSS � �

6. �BCD � �BAD 6. CPCTC

6-5 Conditions for Special ParallelogramsDetermine if the conclusion is valid. If not, tell what additional information is needed to make it valid.

8. Given: _

WY � _

XZ , _

WZ � _

WX Conclusion: WXYZ is a rhombus.

Not enough information. Need to know that WXYZ is a parallelogram.

9. Given: _

WX � _

ZY , _

WZ � _

XY , _

WZ � _

ZY Conclusion: WXYZ is a rectangle.

Valid conclusion

Ready to Go On? Quiz6B

SECTION

A

E

D

B C

8x + 5

12x – 8M L

N

J K

E

C

A B

D

W X

Z Y





6BSECTION

2 4 6

–4

–6

–2

2

4

6

–2–4–6 x

yUse the diagonals to determine whether a parallelogram with the given vertices is a rectangle, a rhombus, or a square. Give all the names that apply.

10. H(3, 5), I(�1, 2), J(�3, �4), K(1, �1)

HIJK is a rhombus.

11. P(2, 4), Q(4, �2), R(�2, �4), S(�4, 2)

PQRS is a rectangle and a square.

12. Given: �MON is equilateral. M is the midpoint of _

LN . LMOP is a parallelogram. Prove: LMOP is a rhombus.

Statements Reasons

1. �MON is equilateral. LMOP is a parallelogram.

1. Given

2. _

OM � _

MN 2. Def. of equilateral �

3. _

LM � _

MN 3. Def. of midpt.

4. _

LM � _

OM 4. Trans. Prop. Of �

5. LMOP is a rhombus. 5. Parallelogram w/ one pair cons. sides � rhombus

6-6 Properties of Kites and TrapezoidsIn kite CDEF, m�CDF � 39�, and m�EFC � 25�. Find each measure.

13. m�CFG 12.5� 14. m�GEF 77.5�

15. m�DCG 51� 16. m�DEF 128.5�

17. Find m�Q. 126� 18. AC � 91.7 and BE � 33.9. Find ED. 57.8

54°

Q

P S

R

E

A B

CD

19. The face of a stone wall is in the shape of a trapezoid. The bases of the wall are 132 in. and 64 in. A steel bar is attached between the midpoint of each leg of the trapezoid. How long is

the bar? 98 in.

P

L M N

O

E

D

C

G

F



Other Special Quadrilaterals

1. Given: ABCD is a rectangle. E, F, G, and H are midpoints of their respective sides. Write a paragraph proof to show that EFGH is a rhombus.

Sample answer: Since ABCD is a rectangle,

_

AD � _

BC and _

AB � _

DC . AD � BC and AB � DC

by the definition of congruent segments. By the

definition of a midpoint, _

AH � _

HD and _

FB � _

FC

and AH � HD and FB � FC. By the Transitive

Property of Equality, AH � HD � FB � FC.

Similarly, by the definition of a midpoint,

_

AE � _

EB and _

DG � _

CG , and AE � EB and

DG � CG. By the Transitive Property of Equality,

AE � EB � DG � CG. Since ABCD is a rectangle,

�A, �B, �C, and �D are right angles.

�AHE � �BFE � �CFG � �DHG. By CPTCT,

_

HE � _

FE � _

FG � _

HG . EFGH is a rhombus

because it is a quadrilateral with four congruent

sides.

2. The quadrilateral at right is a kite, not drawn to scale. Which two sides are congruent,

_ RS and

_ RQ or

_ RS

and _

ST ? Why?

Sample answer: If RS � ST, 4x � 31 � 2x � 35

and x � 2. Substituting x � 2 into the side

lengths, RS � ST � 39 and RQ � �1, which is

impossible. If RS � RQ, 5 � 3x � 2x � 35 and

x � �6. Substituting x � �6 into the side

lengths, RQ � RS � 23 and ST � 7. Therefore,

_

RS � _

RQ .


SECTION

A

E

D

G

B F C

H

Q T

4x + 315 – 3x

2x + 35R S



Writing RatiosWrite a ratio expressing the slope of line t.

slope � rise ____ run � y2 � y1

_________ x2 � x1

� 1 � �2 ___________

2 � (�3) Substitute the given values.

� 3 _____

5 Simplify.

Solving ProportionsSolve each proportion.

A. 15 ___ b � 55 ___ 88

15( 88 ) � b( 55 ) Apply the Cross Products Property.

1320 � 55b Simplify.

1320 _______

55 � 55b ____

55 Divide both sides by 55.

24 � b Solve for b.

B. 27 _____ t � 7 � t � 7 _____ 3

27( 3 ) � ( t � 7 )2 Apply the Cross Products Property.

81 � (t � 7)2 Simplify.

��

81 � ��

� t � 7 � 2

�9 � t � 7 Find the square root of both sides.

t � 7 � 9 or t � 7 � �9 Rewrite as two equations.

t � 16 or t � �2 Solve for t.

Ready To Go On? Skills Intervention7-1 Ratio and Proportion

Vocabulary

ratio proportion extremes means cross products


SECTION

7A

t

2 4

x

y

–4

–2

2

4

–2–4


A proportion is an equation stating that two ratios are equal.

An architect builds a scale model of an office building. The width of the model is 1.5 ft and the height is 2.4 ft. The actual building is 192 ft tall. What is the width of the building?


1. What numbers are being compared in this problem? The widths and heights

of a scale model building and the actual building it represents.

2. What is the problem asking you to find? The width of the actual building.

Make a Plan

3. Let x represent the width of the actual building. Write a proportion that compares the ratios of the height to the width.

width of model building _________________________

height of model building � width of actual building

_________________________ height of actual building

4. Substitute the known values into the proportion: 1.5 ______ 2.4

� x ____ 192

Solve

5. Solve the proportion. 1.5 ______ 2.4

� x ____ 192

1.5 ( 192 ) � ( 2.4 ) x

288 � 2.4x

120 � x

6. The width of the actual building is 120 ft .

Look Back

7. Substitute your value for x into the proportion in Exercise 4. Check to see if both ratios are equal when simplified.

1.5 ___ 2.4 � 5 __ 8 , and 120 ____ 192 � 5 __ 8

8. Are the ratios equal? Yes Is your answer correct? Yes

7AReady To Go On? Problem Solving Intervention7-1 Ratio and Proportion


SECTION


SECTION Ready to Go On? Skills Intervention 6A 6-1 .... 6...Finding Interior Angle Measures and Sums...

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Transcript of SECTION Ready to Go On? Skills Intervention 6A 6-1 .... 6...Finding Interior Angle Measures and Sums...