Download Chapter 6 Notes

6-1
The Polygon Angle-Sum
Theorems
Vocabulary
Review
1. Underline the correct word to complete the sentence.
In a convex polygon, no point on the lines containing the sides of the polygon is in the
interior / exterior of the polygon.
2. Cross out the polygon that is NOT convex.
Vocabulary Builder
REG
yuh lur PAHL ih gahn
Definition: A regular polygon is a polygon that is both equilateral and equiangular.
Example: An equilateral triangle is a regular polygon with three congruent sides
and three congruent angles.
Use Your Vocabulary
Underline the correct word(s) to complete each sentence.
3. The sides of a regular polygon are congruent / scalene .
4. A right triangle is / is not a regular polygon.
5. An isosceles triangle is / is not always a regular polygon.
Write equiangular, equilateral, or regular to identify each hexagon. Use each
word once.
6.
7.
120í
120í
120í
8.
120í
120í
120í
120í
120í
Chapter 6
146
120í
120í
120í
120í
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regular polygon (noun)
Theorem 6-1 Polygon Angle-Sum Theorem and Corollary
Theorem 6-1 The sum of the measures of the interior angles of an n-gon is (n 2 2)180.
Corollary The measure of each interior angle of a regular n-gon is
(n 2 2)180
.
n
9. When n 2 2 5 1, the polygon is a(n) 9.
10. When n 2 2 5 2, the polygon is a(n) 9.
Problem 1 Finding a Polygon Angle Sum
Got It? What is the sum of the interior angle measures of a 17-gon?
11. Use the justifications below to find the sum.
sum 5 Q
5 Q
5
2 2 R 180
12. Draw diagonals from vertex A
to check your answer.
Polygon Angle-Sum Theorem
2 2 R 180
A
Substitute for n.
? 180
Subtract.
Simplify.
5
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13. The sum of the interior angle measures of a 17-gon is
.
Problem 2 Using the Polygon Angle-Sum Theorem
Got It? What is the measure of each interior angle in a regular nonagon?
Underline the correct word or number to complete each sentence.
14. The interior angles in a regular polygon are congruent / different .
15. A regular nonagon has 7 / 8 / 9 congruent sides.
16. Use the Corollary to the Polygon Angle-Sum Theorem to find the measure of each
interior angle in a regular nonagon.
Q
Measure of an angle 5
5
Q
2 2 R 180
R 180
5
17. The measure of each interior angle in a regular nonagon is
147
.
Lesson 6-1
Problem 3 Using the Polygon Angle-Sum Theorem
Got It? What is mlG in quadrilateral EFGH?
G
18. Use the Polygon Angle-Sum Theorem to find m/G for n 5 4.
F 120
m/E 1 m/F 1 m/G 1 m/H 5 (n 2 2)180
m/E 1 m/F 1 m/G 1 m/H 5 Q
1
1
1
5
m/G 1
5
85
E
2 2 R 180
53
H
? 180
m/G 5
19. m/G in quadrilateral EFGH is
.
Theorem 6-2 Polygon Exterior Angle-Sum Theorem
The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.
20. In the pentagon below, m/1 1 m/2 1 m/3 1 m/4 1 m/5 5
3
.
2
4
1
5
21.
120í
22.
81í
90í
56í
75í
87í
72í
120 1 81 1
66í
1 87 5 360
90 1
73í
1 75 1 73 1 66 5
Problem 4 Finding an Exterior Angle Measure
Got It? What is the measure of an exterior angle of a regular nonagon?
Underline the correct number or word to complete each sentence.
23. Since the nonagon is regular, its interior angles are congruent / right .
24. The exterior angles are complements / supplements of the interior angles.
25. Since the nonagon is regular, its exterior angles are congruent / right .
26. The sum of the measures of the exterior angles of a polygon is 180 / 360 .
27. A regular nonagon has 7 / 9 / 12 sides.
Chapter 6
148
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Use the Polygon Exterior Angle-Sum Theorem to find each measure.
28. What is the measure of an exterior angle of a regular nonagon? Explain.
_______________________________________________________________________
_______________________________________________________________________
Lesson Check • Do you UNDERSTAND?
Error Analysis Your friend says that she measured an interior angle of a regular polygon as
130. Explain why this result is impossible.
29. Use indirect reasoning to find a contradiction.
Assume temporarily that a regular n-gon has a 1308 interior angle.
?n
angle sum 5
angle sum 5 Q
5 Q
5
A regular n-gon has n congruent angles.
R 180
Polygon Angle-Sum Theorem
2
Use the Distributive Property.
R 180
Use the Transitive Property of Equality.
Subtract 180n from each side.
5
n5
Divide each side by 250.
n2
The number of sides in a polygon is a whole number $ 3.
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30. Explain why your friend’s result is impossible.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
equilateral polygon
equiangular polygon
regular polygon
Rate how well you can find angle measures of polygons.
Need to
review
0
2
4
6
8
Now I
get it!
10
149
Lesson 6-1
Name
6-1
Class
Date
Additional Problems
The Polygon Angle-Sum Theorems
Problem 1
What is the sum of the angle measures of a 10-gon?
Problem 2
Marcy creates a floor tile pattern using
squares, regular hexagons, and regular
dodecagons (12-sided polygons). What
is the measure of each angle in one
regular dodecagon?
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69
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6-1
Class
Date
Additional Problems (continued)
The Polygon Angle-Sum Theorems
Problem 3
What is m/D in quadrilateral ABCD?
A
98
B
91
112
C
D
Problem 4
What is the measure of an exterior angle
of a regular hexagon?
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1
Name
6-1
Class
Date
Practice
Form G
The Polygon Angle-Sum Theorems
Find the sum of the angle measures of each polygon.
1.
2.
3.
4. 12-gon
5. 18-gon
6. 25-gon
7. 60-gon
8. 102-gon
9. 17-gon
10. 36-gon
11. 90-gon
12. 11-gon
Find the measure of one angle in each regular polygon. Round to the nearest
tenth if necessary.
13.
14.
15.
16. regular 15-gon
17. regular 11-gon
18. regular 13-gon
19. regular 24-gon
20. regular 360-gon
21. regular 18-gon
22. regular 36-gon
23. regular 72-gon
24. regular 144-gon
Algebra Find the missing angle measures.
25.
27.
26.
28.
29.
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Class
Date
Practice (continued)
Form G
The Polygon Angle-Sum Theorems
Algebra Find the missing angle measures.
30.
31.
32.
33.
34.
35.
Find the measure of an exterior angle of each regular polygon. Round to the
nearest tenth if necessary.
36. decagon
37. 16-gon
38. hexagon
39. 20-gon
40. 72-gon
41. square
42. 15-gon
43. 25-gon
44. 80-gon
Find the values of the variables for each regular polygon. Round to the nearest
tenth if necessary.
45.
46.
47.
48. Reasoning Can a quadrilateral have no obtuse angles? Explain.
The measure of an exterior angle of a regular polygon is given. Find the measure of
an interior angle. Then find the number of sides.
49. 12
50. 6
51. 45
52. 40
53. 24
54. 18
55. 9
56. 14.4
57. 7.2
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Class
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Date
Practice
Form K
The Polygon Angle-Sum Theorems
Find the sum of the angle measures of each polygon.
To start, determine the sum of the angles using
the Polygon Angle-Sum Theorem.
1.
Sum = (n − 2)180
=(
– 2)180 =
2. 21-gon
3. 42-gon
4. 50-gon
5. 205-gon
Find the measure of one angle in each regular polygon.
To start, write the formula used to calculate the
measure of an angle of a regular polygon. Then
substitute n = 9 into the formula.
6.
(n − 2)180 (
=
n
− 2)180
.
=
8.
Find the missing angle measures.
9. To start, determine the sum of the angles
using the Polygon Angle-Sum Theorem.
Sum = (n − 2)180 = (6 − 2)180 = 720
Write an equation relating each interior
angle to the sum of the angles.
n + 156 + 122 + 143 + 108 + 110 = 720
10.
11.
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5
Name
Class
6-1
Practice
Date
(continued)
Form K
The Polygon Angle-Sum Theorems
Find the measure of an exterior angle of each regular polygon.
12. 12-gon
13. 24-gon
14. 45-gon
The sum of the angle measures of a polygon with n sides is given. Find n.
15. 900
16. 1440
17. 2340
18. Carly built a Ferris wheel using her construction toys. The frame of the wheel is a
regular 16-gon. Find the sum of the angle measures of the Ferris wheel and the
measure of one angle.
Algebra Find the value of each variable.
19.
20.
21. Your friend wants to build the picture frame shown at the right.
a. What regular polygon is the inside of the frame?
b. Find the measure of each numbered angle.
c. Reasoning If you extended one of the exterior sides of the outside of the
frame, would the measure of the exterior angle
be the same as the measure of ∠2? Explain.
22. Caning chair seats first became popular in
England in the 1600s. This method of weaving
natural materials produces a pattern that
contains several polygons. Identify the outlined
polygon. Then, assuming that the polygon is
regular, find the measure of each numbered
angle.
23. Algebra The measure of an interior angle of a regular polygon is four
times the measure of an exterior angle of the same polygon. What is the
name of the polygon?
24. Reasoning The measure of the exterior angle of a regular polygon is 30.
What is the measure of an interior angle of the same polygon? Explain.
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6
Name
Class
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Date
ELL Support
The Polygon Angle-Sum Theorems
Use a word from the list below to complete each sentence.
concave
convex
equiangular polygon
equilateral polygon
exterior angle
interior angle
regular polygon
1. A polygon that has an interior angle greater than 180° is a _________________ polygon.
2. A polygon that has no interior angles greater than 180° is a _________________
polygon.
3. A hexagon in which all angles measure 120° is an example of an _______________ .
4. An octagon in which all angles measure 135° and all sides are 6 cm long is an example
of a _________________ .
5. An angle inside a polygon is an ______________ .
Circle the term that applies to the diagram.
6.
equiangular
equilateral
regular
7.
equiangular
equilateral
regular
8.
equiangular
equilateral
regular
Multiple Choice
9. What type of angle is the angle labeled x°?
acute
interior
exterior
straight
10. Which figure is equiangular and equilateral?
circle
rectangle
rhombus
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1
square
6-2
Properties of
Parallelograms
Vocabulary
Review
1. Supplementary angles are two angles whose measures sum to
.
2. Suppose /X and /Y are supplementary. If m/X 5 75, then m/Y 5
.
Underline the correct word to complete each sentence.
B
A 60í
3. A linear pair is complementary / supplementary .
E
4. /AFB and /EFD at the right are complementary / supplementary .
C
F
120í
D
Vocabulary Builder
consecutive (adjective) kun SEK yoo tiv
Definition: Consecutive items follow one after another in uninterrupted order.
Examples: The numbers 23, 22, 21, 0, 1, 2, 3, . . . are consecutive integers.
Non-Example: The letters A, B, C, F, P, . . . are NOT consecutive letters of
the alphabet.
Use Your Vocabulary
Use the diagram at the right. Draw a line from each angle in Column A
to a consecutive angle in Column B.
Column A
/F
6. /C
/E
7. /D
/D
,
9. December, November, October, September,
Chapter 6
C
E
Write the next two consecutive months in each sequence.
8. January, February, March, April,
,
150
B
F
Column B
5. /A
A
D
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Math Usage: Consecutive angles of a polygon share a common side.
Theorems 6-3, 6-4, 6-5, 6-6
Theorem 6-3 If a quadrilateral is a parallelogram, then its opposite sides are congruent.
Theorem 6-4 If a quadrilateral is a parallelogram, then its consecutive angles are
supplementary.
Theorem 6-5 If a quadrilateral is a parallelogram, then its opposite angles are congruent.
Theorem 6-6 If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Use the diagram at the right for Exercises 10–12.
A
D
E
10. Mark parallelogram ABCD to model Theorem 6-3 and Theorem 6-5.
11. AE >
12. BE >
B
C
Problem 1 Using Consecutive Angles
Q
Got It? Suppose you adjust the lamp so that mlS is 86. What is mlR
in ~PQRS?
P
Underline the correct word or number to complete each statement.
R
13. /R and /S are adjacent / consecutive angles, so they are supplementary.
64
S
14. m/R 1 m/S 5 90 / 180
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15. Now find m/R.
16. m/R 5
.
Problem 2 Using Properties of Parallelograms in a Proof
Got It? Use the diagram at the right.
Given: ~ABCD, AK > MK
K
Prove: /BCD > /CMD
17. Circle the classification of nAKM .
equilateral
isosceles
right
B
A
C
M
D
18. Complete the proof. The reasons are given.
Statements
1)
AK >
Reasons
1) Given
2) /DAB >
2) Angles opposite congruent sides of a triangle are congruent.
3) /BCD >
3) Opposite angles of a parallelogram are congruent.
4) /BCD >
4) Transitive Property of Congruence
151
Lesson 6-2
Problem 3 Using Algebra to Find Lengths
Got It? Find the values of x and y in ~PQRS at the right. What are PR and SQ?
P
3y 19. Circle the reason PT > TR and ST > TQ.
Diagonals of a
parallelogram
bisect each other.
1
PR is the
perpendicular
bisector of QS.
Opposite sides of
a parallelogram
are congruent.
S
Q
7
x T
y
2x
R
20. Cross out the equation that is NOT true.
3(x 1 1) 2 7 5 2x
y5x11
3y 2 7 5 x 1 1
3y 2 7 5 2x
21. Find the value of x.
22. Find the value of y.
23. Find PT.
24. Find ST.
PT 5 3
PT 5
ST 5
27
11
ST 5
27
PT 5
PR 5 2(
26. Find SQ.
SQ 5 2(
)
PR 5
)
SQ 5
27. Explain why you do not need to find TR and TQ after finding PT and ST.
_______________________________________________________________________
_______________________________________________________________________
Theorem 6-7
If three (or more) parallel lines cut off congruent segments on one transversal, then
they cut off congruent segments on every transversal.
Use the diagram at the right for Exercises 28 and 29.
* ) * ) * )
28. If AB 6 CD 6 EF and AC > CE, then BD >
A
.
29. Mark the diagram to show your answer to Exercise 28.
Chapter 6
152
C
E
B
D
F
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25. Find PR.
Problem 4 Using Parallel lines and Transversals
* ) * ) * ) * )
Got It? In the figure at the right, AE n BF n CG n DH . If EF 5 FG 5 GH 5 6
30. You know that the parallel lines cut off congruent segments on transversal
31. By Theorem 6-7, the parallel lines also cut off congruent segments on
32. AD 5 AB 1 BC 1
33. AB 5
B
A
and AD 5 15, what is CD?
.
C
E
F
.
G
H
by the Segment Addition Postulate.
5 CD, so AD 5
? CD. Then CD 5
34. You know that AD 5 15, so CD 5
? 15 5
D
? AD.
.
Lesson Check • Do you UNDERSTAND?
Error Analysis Your classmate says that QV 5 10. Explain why the
statement may not be correct.
P
Q
S 5 cm
R
T
35. Place a ✓ in the box if you are given the information. Place an ✗ if you
are not given the information.
V
three lines cut by two transversals
three parallel lines cut by two transversals
congruent segments on one transversal
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36. What needs to be true for QV to equal 10?
_______________________________________________________________________
37. Explain why your classmate’s statement may not be correct.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
parallelogram
opposite sides
opposite angles
consecutive angles
Rate how well you understand parallelograms.
Need to
review
0
2
4
6
8
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get it!
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Lesson 6-2
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6-2
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Date
Additional Problems
Properties of Parallelograms
Problem 1
What is m/R in ~RSTU ?
R
U
A. 109
B. 99
S
C. 81
71
T
D. 71
Problem 2
Given: ~ABCD and ~NMLB
A
Prove: /M > /D
D
N
B
M
L
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71
C
Name
Class
Date
Additional Problems (continued)
6-2
Properties of Parallelograms
Problem 3
Solve a system of linear equations to
find the values of a and b in ~HIJK .
What are HJ and IK?
H
K
a
2
2b
a
b
8
I
J
Problem 4
In the figure below, RW 6 SV and SV 6 TU . If RS 5 ST 5 5
and WV 5 7, what is WU?
R
S
T
W
V
U
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Practice
6-2
Form G
Properties of Parallelograms
Find the value of x in each parallelogram.
1.
2.
3.
4.
5.
6.
Developing Proof Complete this two-column proof.
7. Given:
EFGH, with diagonals EG and
HF
Prove: ∆EFK ≅ ∆GHK
Statements
Reasons
2)
1) Given
2) The diagonals of a parallelogram
bisect each other.
3) EF ≅ GH
3)
4)
4)
1)
Algebra Find the values for x and y in
ABCD.
8. AE = 3x, EC = y, DE = 4x, EB = y + 1
9. AE = x + 5, EC = y, DE = 2x + 3, EB = y + 2
10. AE = 3x, EC = 2y − 2, DE = 5x, EB = 2y + 2
11. AE = 2x, EC = y + 4, DE = x, EB = 2y − 1
12. AE = 4x, EC = 5y − 2, DE = 2x, EB = y + 14
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Practice (continued)
Form G
Properties of Parallelograms
In the figure, TU = UV. Find each length.
13. NM
14. QR
15. LN
16. QS
Find the measures of the numbered angles for each parallelogram.
17.
18.
19.
20.
21.
22.
23.
24.
25. Developing Proof A rhombus is a parallelogram with four congruent
sides. Write a plan for the following proof that uses SSS and a property of
parallelograms.
Given: Rhombus ABCD with diagonals AC and
BD intersecting at E
Prove: AC ⊥ BD
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ELL Support
Properties of Parallelograms
Concept List
congruent angles
congruent segments
consecutive angles
diagonal
is parallel to
opposite angles
opposite sides
parallelogram
transversal
Choose the concept from the list above that best represents the item in each box.
1.
∠A and ∠B
4. EF
7.
and HG
WX = YZ
2. P
5.
3.
∠H and ∠F
6.
m∠X = m∠Y
9.
8. Y EFGH
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11
6-3
Proving That a Quadrilateral
Is a Parallelogram
Vocabulary
Review
1. Does a pentagon have opposite sides?
Yes / No
2. Does an n-gon have opposite sides if n is an odd number?
Yes / No
Draw a line from each side in Column A to the opposite side in Column B.
Column A
Column B
3. AB
BC
4. AD
DC
A
B
D
C
Vocabulary Builder
parallelogram
S
P
Definition: A parallelogram is a quadrilateral with two pairs
of opposite sides parallel. Opposite sides may include arrows
to show the sides are parallel.
R
Q
Related Words: square, rectangle, rhombus
Use Your Vocabulary
Write P if the statement describes a parallelogram or NP if it does not.
5. octagon
6. five congruent sides
7. regular quadrilateral
Write P if the figure appears to be a parallelogram or NP if it does not.
8.
Chapter 6
9.
10.
154
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parallelogram (noun) pa ruh LEL uh gram
Theorems 6-8 through 6-12
Theorem 6-8 If both pairs of opposite sides of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Theorem 6-9 If an angle of a quadrilateral is supplementary to both of its consecutive angles,
then the quadrilateral is a parallelogram.
Theorem 6-10 If both pairs of opposite angles of a quadrilateral are congruent, then
the quadrilateral is a parallelogram.
Theorem 6-11 If the diagonals of a quadrilateral bisect each other, then the quadrilateral
is a parallelogram.
Theorem 6-12 If one pair of opposite sides of a quadrilateral is both congruent and
parallel, then the quadrilateral is a parallelogram.
B
Use the diagram at the right and Theorems 6-8 through 6–12 for Exercises 11–16.
11. If AB >
, and BC >
12. If m/A 1 m/B 5
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13. If /A > /
and /
, then ABCD is a ~.
and m/
C
1 m/D 5 180, then ABCD is a ~.
> /D, then ABCD is a ~.
14. If AE >
and BE >
, then ABCD is a ~.
15. If BC >
and BC 6
, then ABCD is a ~.
16. If CD >
and CD 6
, then ABCD is a ~.
A
D
Problem 1 Finding Values for Parallelograms
Got It? Use the diagram at the right. For what values of x and y must EFGH
E
be a parallelogram?
17. Circle the equation you can use to find the value of y. Underline the equation
you can use to find the value of x.
y 1 10 5 3y 2 2
y 1 10 5 4x 1 13
18. Find y.
H
(3y 2) (4x 13) F
(y 10) (12x 7) G
( y 1 10) 1 (3y 2 2) 5 180
19. Find x.
20. What equation could you use to find the value of x first?
21. EFGH must be a parallelogram for x 5
and y 5
155
.
Lesson 6-3
Problem 2
Deciding Whether a Quadrilateral Is a Parallelogram
Got It? Can you prove that the quadrilateral is a parallelogram based
D
E
on the given information? Explain.
Given: EF > GD, DE 6 FG
G
Prove: DEFG is a parallelogram.
F
22. Circle the angles that are consecutive with /G.
/D
/E
/F
23. Underline the correct word to complete the sentence.
Same-side interior angles formed by parallel lines cut by a transversal are
complementary / congruent / supplementary .
24. Circle the interior angles on the same side of transversal DG. Underline the interior
angles on the same side of transversal EF .
/D
/E
/F
/G
25. Can you prove DEFG is a parallelogram? Explain.
______________________________________________________________________________
______________________________________________________________________________
Problem 3 Identifying Parallelograms
raise the platform. What is the maximum height that the vehicle lift can elevate the
truck? Explain.
Q
Q
R
R
26 ft
6 ft
P
26 ft
6 ft
6 ft
26 ft
S
6 ft
26 ft
P
26. Do the lengths of the opposite sides change as the truck is lifted?
27. The least and greatest possible angle measures for /P and /Q are
28. The greatest possible height is when m/P and m/Q are
S
Yes / No
and
.
.
29. What is the maximum height that the vehicle lift can elevate the truck? Explain.
______________________________________________________________________________
______________________________________________________________________________
Chapter 6
156
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Got It? Reasoning A truck sits on the platform of a vehicle lift. Two moving arms
Lesson Check • Do you UNDERSTAND?
Compare and Contrast How is Theorem 6-11 in this lesson different from Theorem 6-6
in the previous lesson? In what situations should you use each theorem? Explain.
For each theorem, circle the hypothesis and underline the conclusion.
30. Theorem 6-6
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
31. Theorem 6-11
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Draw a line from each statement in Column A to the corresponding diagram in
Column B.
Column A
Column B
32. A quadrilateral is a parallelogram.
33. The diagonals of a quadrilateral
bisect each other.
34. Circle the word that describes how Theorem 6-6 and Theorem 6-11 are related.
contrapositive
converse
inverse
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35. In which situations should you use each theorem? Explain.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
diagonal
parallelogram
quadrilateral
Rate how well you can prove that a quadrilateral is a parallelogram.
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Name
6-3
Class
Date
Additional Problems
Proving That a Quadrilateral is a Parallelogram
Problem 1
For what value of x must RSTU
be a parallelogram?
R
x8
U
4y 3
2y 5
S
T
5y
Problem 2
a. Can you prove the quadrilateral is
D
A
a parallelogram based on the given
information? What theorem can you use?
Given: AE 5 CE 5 14, DB 5 2DE
E
B
Prove: ABCD is a parallelogram
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ELL Support
Proving That a Quadrilateral Is a Parallelogram
The converse of a statement reverses the conclusion and the hypothesis.
Example: If A is true, then B is true.
Converse: If B is true, then A is true.
Hypothesis: A is true.
Hypothesis: B is true.
Conclusion: B is true.
Conclusion: A is true.
Sample
Example: If a number is 5 more than 7,
then the number is 12.
Converse: If a number is 12, then the
number is 5 more than 7.
For each statement below, circle the hypothesis and underline the conclusion.
Then write the converse.
1. If an apple is red, then the apple is ripe. _______________________________
2. If the tree has leaves, then the season is summer.
______________________________________________________________
3. Complete the converse of this statement: If water is solid, then it is frozen.
Converse: If _______________ , then _______________ .
The converse of a theorem reverses the conclusion and the hypothesis.
Sample
Theorem: If a transversal intersects two
parallel lines, then corresponding angles
are congruent.
Converse: If two lines and a transversal
form alternate interior angles that are
congruent, then the two lines are parallel.
Match each theorem from Section A with its converse in Section B.
Section A:
3. If a quadrilateral is a parallelogram,
then both pairs of opposite sides are
congruent.
Section B:
If the diagonals of a quadrilateral bisect
each other, then the quadrilateral is a
parallelogram.
4.
If a quadrilateral is a parallelogram,
then its diagonals bisect each other.
If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
If a quadrilateral is a parallelogram,
then both pairs of opposite angles are
congruent.
If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
5.
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21
6-4
Properties of Rhombuses,
Rectangles, and Squares
Vocabulary
Review
1. Circle the segments that are diagonals.
A
AG
AC
HD
GC
H
BF
AE
EG
EF
G
2. Is a diagonal ever a line or a ray?
3. The diagonals of quadrilateral JKLM are
B
C
D
F
E
Yes / No
and
.
Vocabulary Builder
rhombus
rhombus (noun)
RAHM
bus
Main Idea: A rhombus has four congruent sides but not necessarily
four right angles.
Examples: diamond, square
Use Your Vocabulary
Complete each statement with always, sometimes, or never.
4. A rhombus is 9 a parallelogram.
5. A parallelogram is 9 a rhombus.
6. A rectangle is 9 a rhombus.
7. A square is 9 a rhombus.
8. A rhombus is 9 a square.
9. A rhombus is 9 a hexagon.
Chapter 6
158
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Definition: A rhombus is a parallelogram with four congruent sides.
Key Concept Special Parallelograms
A rhombus is a parallelogram
with four congruent sides.
A rectangle is a parallelogram
with four right angles.
A square is a parallelogram
with four congruent sides
and four right angles.
10. Write the words rectangles, rhombuses, and squares in the Venn diagram below to
show that one special parallelogram has the properties of the other two.
Special Parallelograms
Problem 1 Classifying Special Parallograms
Got It? Is ~EFGH a rhombus, a rectangle, or a square? Explain.
E
11. Circle the number of sides marked congruent in the diagram.
1
2
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
12. Are any of the angles right angles?
3
H
4
F
Yes / No
G
13. Is ~EFGH a rhombus, a rectangle, or a square? Explain.
_______________________________________________________________________
_______________________________________________________________________
Theorems 6-13 and 6-14
Theorem 6-13 If a parallelogram is a rhombus, then its diagonals are perpendicular.
Theorem 6-14 If a parallelogram is a rhombus, then each diagonal bisects a pair
of opposite angles.
A
Use the diagram at the right for Exercises 14–18.
14. If ABCD is a rhombus, then AC '
.
15. If ABCD is a rhombus, then AC bisects /
and /
16. If ABCD is a rhombus, then /1 > /2 > /
>/
17. If ABCD is a rhombus, then BD bisects /
and /
18. If ABCD is a rhombus, then /3 > /
D
3
1 2
> /
159
> /
.
.
8
B
7
6
4
5
C
.
.
Lesson 6-4
Finding Angle Measures
Problem 2
Got It? What are the measures of the numbered angles in rhombus PQRS?
Q
104
19. Circle the word that describes nPQR and nRSP.
equilateral
isosceles
right
1
20. Circle the congruent angles in nPQR. Underline the congruent angles in nRSP.
/1
/2
/3
/4
21. m/1 1 m/2 1 104 5
/Q
P
3
R
4
2
S
/S
22. m/1 1 m/2 5
23. m/1 5
24. Each diagonal of a rhombus 9 a pair of opposite angles.
25. Circle the angles in rhombus PQRS that are congruent.
/1
26. m/1 5
/2
, m/2 5
/3
, m/3 5
/4
, and m/4 5
.
Theorem 6-15
Theorem 6-15 If a parallelogram is a rectangle, then its diagonals are congruent.
27. If RSTU is a rectangle, then RT >
.
Got It? If LN 5 4x 2 17 and MO 5 2x 1 13 , what are the lengths of the
N
M
diagonals of rectangle LMNO?
Underline the correct word to complete each sentence.
P
28. LMNO is a rectangle / rhombus .
L
29. The diagonals of this figure are congruent / parallel .
30. Complete.
LN 5
, so 4x 2 17 5
31. Write and solve an equation to find the
value of x.
.
32. Use the value of x to find the length
of LN.
33. The diagonals of a rectangle are congruent, so the length of each diagonal is
Chapter 6
160
.
O
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Problem 3 Finding Diagonal Length
Lesson Check • Do you UNDERSTAND?
Error Analysis Your class needs to find the value of x for which ~DEFG
is a rectangle. A classmate’s work is shown below. What is the error? Explain.
G
D
2x + 8 = 9x - 6
(9x 6)
14 = 7x
2=x
E
F
(2x 8)
Write T for true or F for false.
34. If a parallelogram is a rectangle, then each diagonal bisects a pair of opposite
angles.
35. If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite
angles.
36. If DEFG is a rectangle, m/D 5 m/
37. m/F 5
5 m/
5 m/
.
.
38. What is the error? Explain.
_______________________________________________________________________
_______________________________________________________________________
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_______________________________________________________________________
39. Find the value of x for which ~DEFG is a rectangle.
40. The value of x for which ~DEFG is a rectangle is
.
Math Success
Check off the vocabulary words that you understand.
parallelogram
rhombus
rectangle
square
diagonal
Rate how well you can find angles and diagonals of special parallelograms.
Need to
review
0
2
4
6
8
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get it!
10
161
Lesson 6-4
Name
6-4
Class
Date
Additional Problems
Properties of Rhombuses, Rectangles, and Squares
Problem 1
Is parallelogram FGHJ a rhombus,
a rectangle, or a square? Explain.
G
H
F
J
Problem 2
D
What are the measures of the numbered
angles in rhombus ABCD?
3
A
2
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C
100 1
B
Name
6-4
Class
Date
Additional Problems (continued)
Properties of Rhombuses, Rectangles, and Squares
Problem 3
In rectangle MNOP, PN 5 7x 2 8 and
MO 5 4x 1 10. What is the length
of PN ?
P
O
M
N
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Name
6-4
Class
Date
Practice
Form K
Properties of Rhombuses, Rectangles, and Squares
Decide whether the parallelogram is a rhombus, a rectangle, or a square. Explain
1.
2.
3.
4.
Find the measures of the numbered angles in each rhombus.
5.
To start, a diagonal of a rhombus forms
an isosceles triangle with congruent base
angles.
So, m∠
6.
8.
= 38.
7.
To start, the diagonals of a rhombus are
perpendicular.
So, m∠
9.
= 90.
10.
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Name
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Practice (continued)
Form K
Properties of Rhombuses, Rectangles, and Squares
Algebra QRST is a rectangle. Find the value of x and the length of
each diagonal.
11. QS = x and RT 5 6x − 10
To start, write an equation to show the
diagonals are congruent.
=
12. QS = 4x − 7 and RT = 2x + 11
13. QS = 5x + 12 and RT = 6x − 2
14. QS = 6x − 3 and RT = 4x + 19
15. QS = x + 45 and RT = 4x − 45
Determine the most precise name for each quadrilateral.
16.
17.
18.
19.
Determine whether each statement is true or false. If it is false, rewrite the
sentence to make it true. If it is true, list any other quadrilaterals for which
the sentence would be true.
20. Rhombuses have four congruent sides.
21. Rectangles have four congruent angles.
22. The diagonals of a rectangle bisect the opposite angles.
23. The diagonals of a rhombus are always congruent.
Algebra Find the values of the variables. Then find the side lengths.
24.
25.
Square
Rhombus
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Name
6-4
Class
Date
Practice
Form G
Properties of Rhombuses, Rectangles, and Squares
Decide whether the parallelogram is a rhombus, a rectangle, or a
square. Explain.
1.
2.
3.
4.
5.
Find the measures of the numbered angles in each rhombus.
6.
7.
8.
9.
10.
11.
12.
13.
Algebra HIJK is a rectangle. Find the value of x and the length of each diagonal.
14. HJ = x and IK = 2x −7
15. HJ = 3x + 5 and IK = 5x −9
16. HJ = 3x + 7 and IK = 6x − 11
17. HJ = 19 + 2x and IK = 3x + 22
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Name
6-4
Class
Date
Practice (continued)
Form G
Properties of Rhombuses, Rectangles, and Squares
Algebra HIJK is a rectangle. Find the value of x and the length of each diagonal.
18. HJ = 4x and IK = 7x − 12
19. HJ = x +40 and IK = 5x
Determine the most precise name for each quadrilateral.
20.
21.
21.
23.
Algebra Find the values of the variables. Then find the side lengths.
24. square WXYZ
25. rhombus ABCD
26. rectangle QRST
27. square LMNO
28. Solve using a paragraph proof.
Given: Rectangle DVEO with diagonals DE and OV
Prove: ∆OVE ≅ ∆DEV
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34
Name
Class
6-4
Date
ELL Support
Properties of Rhombuses, Rectangles, and Squares
Use the list below to complete the concept map.
diagonals are perpendicular
parallelogram
square
diagonals are congruent
four right angles
rhombus
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31
6-5
Conditions for Rhombuses,
Rectangles, and Squares
Vocabulary
Review
1. A quadrilateral is a polygon with
sides.
2. Cross out the figure that is NOT a quadrilateral.
Vocabulary Builder
diagonals
diagonal (noun) dy AG uh nul
Word Origin: The word diagonal comes from the Greek prefix dia-,,
which means “through,” and gonia, which means “angle” or “corner.”
Use Your Vocabulary
3. Circle the polygon that has no diagonal.
triangle
quadrilateral
pentagon
hexagon
4. Circle the polygon that has two diagonals.
triangle
quadrilateral
pentagon
hexagon
5. Draw the diagonals from one vertex in each figure.
6. Write the number of diagonals you drew in each of the figures above.
pentagon:
Chapter 6
hexagon:
heptagon:
162
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Definition: A diagonal is a segment with endpoints at two
nonadjacent vertices of a polygon.
Theorems 6-16, 6-17, and 6-18
Theorem 6-16 If the diagonals of a parallelogram are perpendicular, then the
parallelogram is a rhombus.
Theorem 6-17 If one diagonal of a parallelogram bisects a pair of opposite angles,
then the parallelogram is a rhombus.
7. Insert a right angle symbol in the parallelogram at the right to illustrate
Theorem 6-16. Insert congruent marks to illustrate Theorem 6-17.
A
Use the diagram from Exercise 7 to complete Exercises 8 and 9.
8. If ABCD is a parallelogram and AC '
a rhombus.
, then ABCD is
9. If ABCD is a parallelogram, /1 >
then ABCD is a rhombus.
, and /3 >
D
3
1
B
2
4
C
,
Theorem 6-18 If the diagonals of a parallelogram are congruent, then the
parallelogram is a rectangle.
A
D
B
C
10. Insert congruent marks and right angle symbols in the parallelogram to
the right to illustrate Theorem 6-18.
11. Use the diagram from Exercise 10 to complete the statement.
If ABCD is a parallelogram, and BD >
is a rectangle.
then ABCD
12. Circle the parallelogram that has diagonals that are both perpendicular
and congruent.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
parallelogram
rectangle
rhombus
square
Problem 1 Identifying Special Parallelograms
Got It? A parallelogram has angle measures of 20, 160, 20, and 160. Can you
conclude that it is a rhombus, a rectangle, or a square? Explain.
13. Draw a parallelogram in the box below. Label the angles with their measures.
Use a protractor to help you make accurate angle measurements.
163
Lesson 6-5
Underline the correct word or words to complete each sentence.
14. You do / do not know the lengths of the sides of the parallelogram.
15. You do / do not know the lengths of the diagonals.
16. The angles of a rectangle are all acute / obtuse / right angles.
17. The angles of a square are all acute / obtuse / right angles.
18. Can you conclude that the parallelogram is a rhombus, a rectangle, or
a square? Explain.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Using Properties of Special Parallelograms
Got It? For what value of y is ~DEFG a rectangle?
D
19. For ~DEFG to be a parallelogram,
the diagonals must 9 each other.
20. EG 5 2 (
3
5y
21. DF 5 2 (
)
5
)
23. Now write an equation and solve for y.
.
Problem 3 Using Properties of Parallelograms
Got It? Suppose you are on the volunteer building team at the right.
You are helping to lay out a square play area. How can
you use properties of diagonals to locate the four corners?
25. You can cut two pieces of rope that will be the
diagonals of the square play area. Cut them the
same length because a parallelogram is a 9
if the diagonals are congruent.
Chapter 6
4
7y
5
4
E
F
5
22. For ~DEFG to be a rectangle, the diagonals must be 9.
24. ~DEFG is a rectangle for y 5
G
164
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Problem 2
26. You join the two pieces of rope at their midpoints because a
quadrilateral is a 9 if the diagonals bisect each other.
27. You move so the diagonals are perpendicular because a
parallelogram is a 9 if the diagonals are perpendicular.
28. Explain why the polygon is a square when you pull the ropes taut.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Lesson Check • Do you UNDERSTAND?
Name all of the special parallelograms that have each property.
A. Diagonals are perpendicular.
B. Diagonals are congruent.
C. Diagonals are angle bisectors.
D. Diagonals bisect each other.
E. Diagonals are perpendicular bisectors of each other.
29. Place a ✓ in the box if the parallelogram has the property. Place an ✗ if it does not.
Property
Rectangle
Rhombus
Square
A
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B
C
D
E
Math Success
Check off the vocabulary words that you understand.
rhombus
rectangle
square
diagonal
Rate how well you can use properties of parallelograms.
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review
0
2
4
6
8
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get it!
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165
Lesson 6-5
Name
Class
6-5
Date
Additional Problems
Conditions for Rhombuses, Rectangles, and Squares
Problem 1
Can you conclude that the parallelogram is a rhombus, a
rectangle, or a square? Explain.
a.
7
7
b.
90
7
7
Problem 2
For what value of x is parallelogram ABCD
a square?
A
(6x 6)
B
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D
C
Name
6-5
Class
Date
Practice
Form G
Conditions for Rhombuses, Rectangles, and Squares
Can you conclude that the parallelogram is a rhombus, a rectangle, or a
square? Explain.
1.
2.
3.
4.
For what value of x is the figure the given special parallelogram?
5. rhombus
8. rhombus
6. rhombus
7. rectangle
9. rectangle
10. rectangle
Open-Ended Given two segments with lengths x and y (x ≠ y), what special
parallelograms meet the given conditions? Show each sketch.
11. One diagonal has length x, the other has length y. The diagonals intersect at right
angles.
12. Both diagonals have length y and do not intersect at right angles.
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Name
6-5
Class
Date
Practice (continued)
Form G
Conditions for Rhombuses, Rectangles, and Squares
For Exercises 13–16, determine whether the parallelogram is a rhombus, a
rectangle, or a square. Give the most precise description in each case.
13. A parallelogram has perpendicular diagonals and angle measures of 45, 135, 45, and
135.
14. A parallelogram has perpendicular and congruent diagonals.
15. A parallelogram has perpendicular diagonals and angle measures that are all
90.
16. A parallelogram has congruent diagonals.
17. A woman is plotting out a garden bed. She measures the diagonals of the bed and finds
that one is 22 ft long and the other is 23 ft long. Could the garden bed be a rectangle?
Explain.
18. A man is making a square frame. How can he check to make sure the frame is square,
using only a tape measure?
19. A girl cuts out rectangular pieces of cardboard for a project. She checks to see that they
are rectangular by determining if the diagonals are perpendicular. Will this tell her
whether a piece is a rectangle? Explain.
20. Reasoning Explain why drawing both diagonals on any rectangle will always result
in two pairs of nonoverlapping congruent triangles.
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44
6-6
Trapezoids and Kites
Vocabulary
Review
Underline the correct word to complete each sentence.
1. An isosceles triangle always has two / three congruent sides.
2. An equilateral triangle is also a(n) isosceles / right triangle.
3. Cross out the length(s) that can NOT be side lengths of an isosceles triangle.
3, 4, 5
8, 8, 10
3.6, 5, 3.6
7, 11, 11
Vocabulary Builder
trapezoid
TRAP
ih zoyd
base
leg
Related Words: base, leg
Definition: A trapezoid is a quadrilateral with exactly
one pair of parallel sides.
base angles
base
Main Idea: The parallel sides of a trapezoid are called bases. The nonparallel
sides are called legs. The two angles that share a base of a trapezoid are called
base angles.
Use Your Vocabulary
4. Cross out the figure that is NOT a trapezoid.
5. Circle the figure(s) than can be divided into two trapezoids. Then divide each figure
that you circled into two trapezoids.
Chapter 6
166
leg
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trapezoid (noun)
Theorems 6-19, 6-20, and 6-21
Theorem 6-19 If a quadrilateral is an isosceles trapezoid, then each pair of base angles
is congruent.
Theorem 6-20 If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.
6. If TRAP is an isosceles trapezoid with bases RA and TP,
then /T > /
and /R > /
R
A
.
7. Use Theorem 6-19 and your answers to Exercise 6 to draw
congruence marks on the trapezoid at the right.
P
T
8. If ABCD is an isosceles trapezoid, then AC >
B
.
C
9. If ABCD is an isosceles trapezoid and AB 5 5 cm, then
CD 5
cm
cm.
10. Use Theorem 6-20 and your answer to Exercises 8
and 9 to label the diagram at the right.
cm
D
A
Theorem 6-21 Trapezoid Midsegment Theorem If a quadrilateral
is a trapezoid, then
(1) the midsegment is parallel to the bases, and
(2) the length of the midsegment is half the sum of the lengths of the bases.
11. If TRAP is a trapezoid with midsegment MN, then
(2) MN 5 12 Q
6
A
N
M
1
R
P
T
Problem 2 Finding Angle Measures in Isosceles Trapezoids
Got It? A fan has 15 angles meeting at the center. What are the measures of the
base angles of the congruent isosceles trapezoids in its second ring?
dí
Use the diagram at the right for Exercises 12–16.
one
two
13. a 5 360 4
14. b 5
180 2
2
three
cí
12. Circle the number of isosceles triangles in each
wedge. Underline the number of isosceles
trapezoids in each wedge.
bí
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(1) MN 6
R
aí
four
5
5
15. c 5 180 2
5
16. d 5 180 2
5
17. The measures of the base angles of the isosceles trapezoids are
167
and
.
Lesson 6-6
Problem 3 Using the Midsegment Theorem
Q 10 R
Got It? Algebra MN is the midsegment of trapezoid PQRS. What is x?
2x 11
M
What is MN?
P
8x 12
18. The value of x is found below. Write a reason for each step.
N
S
MN 5 12 (QR 1 PS)
2x 1 11 5 12 f10 1 (8x 2 12)g
2x 1 11 5 12 (8x 2 2)
2x 1 11 5 4x 2 1
2x 1 12 5 4x
12 5 2x
65x
19. Use the value of x to find MN.
Theorem 6-22
Theorem 6-22 If a quadrilateral is a kite, then its diagonals are perpendicular.
B
20. If ABCD is a kite, then AC '
.
21. Use Theorem 6-22 and Exercise 20 to draw
congruence marks and right angle symbol(s) on
the kite at the right.
C
A
D
Problem 4 Finding Angle Measures in Kites
Got It? Quadrilateral KLMN is a kite. What are ml1, ml2, and ml3?
22. Diagonals of a kite are perpendicular, so m/1 5
23. nKNM > nKLM by SSS, so m/3 5 m/NKM 5
24. m/2 5 m/1 2 m/
.
168
L
3
K
by the Triangle Exterior Angle Theorem.
25. Solve for m/2.
Chapter 6
.
2
1
36
N
M
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A kite is a quadrilateral with two pairs of consecutive sides congruent and no opposite
sides congruent.
Lesson Check • Do you UNDERSTAND?
Compare and Contrast How is a kite similar to a rhombus? How is it different? Explain.
26. Place a ✓ in the box if the description fits the figure. Place an ✗ if it does not.
Kite
Description
Rhombus
Quadrilateral
Perpendicular diagonals
Each diagonal bisects a pair of opposite angles.
Congruent opposite sides
Two pairs of congruent consecutive sides
Two pairs of congruent opposite angles
Supplementary consecutive angles
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27. How is a kite similar to a rhombus? How is it different? Explain.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
trapezoid
kite
base
leg
midsegment
Rate how well you can use properties of trapezoids and kites.
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review
0
2
4
6
8
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get it!
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Lesson 6-6
Name
Class
Date
Additional Problems
6-6
Trapezoids and Kites
Problem 1
RSTU is an isosceles trapezoid and
m/S 5 75. What are m/R, m/T ,
and m/U ?
R
S
75
U
T
Problem 2
What are the values of x and y in the isosceles triangle below
if DE 6 BC?
D
x
A
32
B
y
E
C
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79
Name
6-6
Class
Date
Additional Problems (continued)
Trapezoids and Kites
Problem 3
TU is the midsegment of trapezoid WXYZ.
What is x?
6x 12
W
2x 10
T
X
18
Z
U
Y
Problem 4
Quadrilateral ABCD is a kite. What are
m/1 and m/2?
A
B
2
40
1
C
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80
D
Name
6-6
Class
Date
ELL Support
Trapezoids and Kites
Complete the vocabulary chart by filling in the missing information.
Word or Word
Phrase
Definition
Picture or Example
trapezoid
A trapezoid is a quadrilateral with one
pair of parallel sides.
legs of a trapezoid
1.
TP and RA
bases of a
trapezoid
2.
TR and PA
isosceles trapezoid
An isosceles trapezoid is a trapezoid
with legs that are congruent.
3.
base angles
4.
∠A and ∠B or
∠C and ∠D
kite
A kite is a quadrilateral with two pairs
of consecutive, congruent sides. In a
kite, no opposite sides are congruent.
5.
midsegment of
a trapezoid
6.
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51
Name
6-6
Class
Date
Practice
Form G
Trapezoids and Kites
Find the measures of the numbered angles in each isosceles trapezoid.
1.
2.
3.
4.
5.
6.
7.
Algebra Find the value(s) of the variable(s) in each isosceles trapezoid.
8.
9.
10.
12.
13.
Find XY in each trapezoid.
11.
Algebra Find the lengths of the segments with variable expressions.
14.
15.
16.
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Name
6-6
17.
Class
Date
Practice (continued)
Form G
Trapezoids and Kites
CD is the midsegment of trapezoid WXYZ.
a. What is the value of x?
b. What is XY?
c. What is WZ?
18. Reasoning The diagonals of a quadrilateral form two acute and two
obtuse angles at their intersection. Is this quadrilateral a kite? Explain.
19. Reasoning The diagonals of a quadrilateral form right angles and its side
lengths are 4, 4, 6, and 6. Could this quadrilateral be a kite? Explain.
Find the measures of the numbered angles in each kite.
20.
23.
21.
22.
24.
25.
Algebra Find the value(s) of the variable(s) in each kite.
26.
27.
28.
For which value of x is each figure a Kite?
29.
30.
Prentice Hall Gold Geometry • Teaching Resources
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54
Name
Class
Date
Chapter 6 Quiz 2
Form G
Lessons 6-7 through 6-9
Do you know HOW?
Determine whether each triangle is scalene, isosceles, or equilateral.
1.
2.
Determine whether the parallelogram is a rhombus, rectangle, square, or
none of these.
3.
4.
5.
What are the coordinates of the vertices of each parallelogram?
6.
7.
Do you UNDERSTAND?
9. Reasoning Describe how you would determine LMNO
is a parallelogram.
8.
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92
6-7
Polygons in the
Coordinate Plane
Vocabulary
Review
1. Draw a line from each item in Column A to the corresponding part of the
coordinate plane in Column B.
Column A
Column B
origin
Quadrant I
y
Quadrant II
Quadrant III
x
Quadrant IV
x-axis
Vocabulary Builder
classify (verb)
KLAS
uh fy
Definition: To classify is to organize by category or type.
Math Usage: You can classify figures by their properties.
Related Words: classification (noun), classified (adjective)
Example: Rectangles, squares, and rhombuses are classified as parallelograms.
Use Your Vocabulary
Complete each statement with the correct word from the list. Use each word only once.
classification
classified
classify
2. Trapezoids are 9 as quadrilaterals.
3. Taxonomy is a system of 9 in biology.
4. Schools 9 children by age.
Chapter 6
170
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y-axis
Key Concept Formulas on the Coordinate Plane
Distance Formula
Midpoint Formula
Mâ1
! 2 Ľx1)2 à(y2 Ľy1)2
d â(x
Formula
When to
Use It
Slope Formula
x1 àx2 , y1 ày2
2
2 2
y Ľy
m â x2 Ľx1
2
1
To determine whether
To determine
To determine whether
rsides are congruent
r diagonals are
rthe coordinates of the
ropposite sides are parallel
r diagonals are perpendicular
r sides are perpendicular
midpoint of a side
rwhether diagonals
congruent
bisect each other
Decide when to use each formula. Write D for Distance Formula,
M for Midpoint Formula, or S for Slope Formula.
5. You want to know whether diagonals bisect each other.
6. You want to find whether opposite sides of a quadrilateral are parallel.
7. You want to know whether sides of a polygon are congruent.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Problem 1 Classifying a Triangle
Got It? kDEF has vertices D(0, 0), E(1, 4), and F(5, 2). Is kDEF scalene,
isosceles, or equilateral?
8. Graph nDEF on the coordinate plane at the right.
y
Use the Distance Formula to find the length of each side.
9. EF 5
5
Å
Ä
5 Ä
10. DE 5
5
Å
Ä
a5 2
1
2
b 1 a2 2
b
4
2
2
x
O
a1 2
2
b 1 a4 2
b
2
11. DF 5
1
5
5 Ä
Å
Ä
2
a5 2
4
2
b 1 a2 2
b
2
1
5 Ä
12. What type of triangle is nDEF? Explain.
_______________________________________________________________________
171
Lesson 6-7
Problem 2
Classifying a Parallelogram
Got It? ~MNPQ has vertices M(0, 1), N(21, 4), P(2, 5), and Q(3, 2). Is ~MNPQ a
rectangle? Explain.
13. Find MP and NQ to determine whether the diagonals MP and NQ are congruent.
MP 5
5
Å
2
a2 2
b 1 a5 2
b
2
NQ 5
1
Ä
5
5 Ä
Å
Ä
2
a3 2
b 1 a2 2
b
2
1
5 Ä
14. Is ~MNPQ a rectangle? Explain.
_______________________________________________________________________
Problem 3 Classifying a Quadrilateral
Got It? An isosceles trapezoid has vertices A(0, 0), B(2, 4), C(6, 4), and
D(8, 0). What special quadrilateral is formed by connecting the midpoints
of the sides of ABCD?
y
15. Draw the trapezoid on the coordinate plane at the right.
16. Find the coordinates of the midpoints of each side.
AB
a
2
01
2
,
01
2
b 5 Q
CD
x
R
,
O
BC
2
4
AD
17. Draw the midpoints on the trapezoid and connect them. Judging by appearance,
what type of special quadrilateral did you draw? Circle the most precise answer.
kite
parallelogram
rhombus
trapezoid
18. To verify your answer to Exercise 17, find the slopes of the segments.
connecting midpoints of AB and BC:
connecting midpoints of BC and CD:
connecting midpoints of CD and AD:
connecting midpoints of AD and AB:
19. Are the slopes of opposite segments equal?
Yes / No
20. Are consecutive segments perpendicular?
Yes / No
21. The special quadrilateral is a 9.
Chapter 6
172
6
8
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4
Lesson Check • Do you UNDERSTAND?
y
Error Analysis A student says that the quadrilateral with vertices
D(1, 2), E(0, 7), F(5, 6), and G(7, 0) is a rhombus because its
diagonals are perpendicular. What is the student’s error?
6
22. Draw DEFG on the coordinate plane at the right.
4
23. Underline the correct words to complete Theorem 6-16.
If the diagonals of a parallelogram / polygon are perpendicular,
2
x
then the parallelogram / polygon is a rhombus.
O
2
4
6
8
24. Check whether DEFG is a parallelogram.
slope of DE:
7 2
slope of FG:
5
0 2
slope of DG:
5
7 2
0 2
slope of EF:
5
7 2
6 2
5
5 2
25. Are both pairs of opposite sides parallel?
Yes / No
26. Find the slope of diagonal DF .
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0 2
27. Find the slope of diagonal EG.
28. Are the diagonals perpendicular?
Yes / No
29. Explain the student’s error.
_______________________________________________________________________
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
distance
midpoint
slope
Rate how well you can classify quadrilaterals in the coordinate plane.
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review
0
2
4
6
8
Now I
get it!
10
173
Lesson 6-7
Name
6-7
Class
Date
Additional Problems
Polygons in the Coordinate Plane
Problem 1
Is nRST scalene, isosceles,
or equilateral?
y
6
4
S
R
x
2
6 4
4
6
2
T
4
6
Problem 2
Is parallelogram ABCD
a rhombus? Explain.
6
y
4
C
B
2
x
6 4
2
A
4
6
4
6
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81
2
D
Name
6-7
Class
Date
Additional Problems (continued)
Polygons in the Coordinate Plane
Problem 3
What is the most precise
classification of the quadrilateral
formed by connecting the
midpoints of the sides of the
isosceles trapezoid?
6
4
O
N
2
x
6 4
L
2
2
4
6
Prentice Hall Geometry • Additional Problems
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82
y
2
4
6
M
6-8
Applying Coordinate
Geometry
Vocabulary
Review
Write T for true or F for false.
1. The vertex of an angle is the endpoint of two rays.
2. When you name angles using three points, the vertex gets named first.
3. A polygon has the same number of sides and vertices.
A
B
4. Circle the vertex of the largest angle in nABC at the right.
C
5. Circle the figure that has the greatest number of vertices.
hexagon
kite
rectangle
trapezoid
Vocabulary Builder
coordinates (noun) koh AWR din its
(Ľ1, 3)
Definition: Coordinates are numbers or letters that
specify the location of an object.
x-coordinate
y-coordinate
Math Usage: The coordinates of a point on a plane are an ordered
d d pair
i off numbers.
b
Main Idea: The first coordinate of an ordered pair is the x-coordinate. The second is
the y-coordinate.
Use Your Vocabulary
Draw a line from each point in Column A to its coordinates
in Column B.
Column A
(21, 23)
7. B
(1, 3)
8. C
(3, 21)
9. D
(23, 1)
Ľ4
C
x
Ľ2 O
B
Ľ2
Ľ4
174
y
2
A
Column B
6. A
Chapter 6
4
2
4
D
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coordinates
Problem 1 Naming Coordinates
Got It? RECT is a rectangle with height a and length 2b.
The y-axis bisects EC and RT. What are the coordinates
of the vertices of RECT?
10. Use the information in the problem to mark
all segments that are congruent to OT.
11. Rectangle RECT has length
so RT 5
12. The coordinates of O are (
coordinates of R are (2
C
x
O
R
,
and RO 5 OT 5
y
E
T
.
, 0), so the coordinates of T are (
, 0), and the
, 0).
13. Rectangle RECT has height a, so TC 5 RE 5
14. The coordinates of C are (
,
.
), so the coordinates of E are (
,
).
15. Why is it helpful that one side of rectangle RECT is on the x-axis and the figure is
centered on the y-axis.
_______________________________________________________________________
_______________________________________________________________________
Problem 2
Using Variable Coordinates
y
B(2a à 2b, 2c)
C (2b, 2c)
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Got It? Reasoning The diagram at the right shows
a general parallelogram with a vertex at the origin and
one side along the x-axis. Explain why the x-coordinate
of B is the sum of 2a and 2b.
16. Complete the diagram.
x
O
A(2a, 0)
17. Complete the reasoning model below.
Write
Think
â
Opposite sides of a parallelogram are congruent.
OA â
The x-coordinate is the sum of the lengths in
The x-coordinate of B is
the brackets.
à
â
.
18. Explain why the x-coordinate of B is the sum of 2a 1 2b.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
175
Lesson 6-8
You can use coordinate geometry and algebra to prove theorems in geometry.
This kind of proof is called a coordinate proof.
Problem 3 Planning a Coordinate Proof
Got It? Plan a coordinate proof of the Triangle Midsegment Theorem
(Theorem 5-1).
19. Underline the correct words to complete Theorem 5-1.
If a segment joins the vertices / midpoints of two sides of a triangle, then the
segment is perpendicular / parallel to the third side, and is half its length.
20. Write the coordinates of the vertices of nABC on the grid below. Use multiples of 2
to name the coordinates.
y
B(
F
,
)
E
x
O C(
,
)
A(
,
)
_______________________________________________________________________
_______________________________________________________________________
22. Complete the Given and Prove.
Given: E is the 9 of AB and F is the 9 of BC.
1
Prove: EF 6 AC, and EF 5 2 AC
23. Circle the formula you need to use to prove EF 6 AC. Underline the formula
you need to use to prove EF 5 12 AC.
Distance Formula
Midpoint Formula
Slope Formula
Underline the correct word to complete each sentence.
24. If the slopes of EF and AC are equal, then EF and AC are congruent / parallel .
25. If you know the lengths of EF and AC, then you can add / compare them.
26. Write three steps you must do before writing the plan for a coordinate proof.
_______________________________________________________________________
_______________________________________________________________________
Chapter 6
176
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21. Reasoning Why should you make the coordinates of A and B multiples of 2?
Lesson Check • Do you UNDERSTAND?
Error Analysis A classmate says the endpoints of the midsegment of the
y
R(2b, 2c)
d1a c
b c
trapezoid at the right are Q 2 , 2 R and Q 2 , 2 R . What is your classmate’s
error? Explain.
27. What is the Midpoint Formula?
M5 a
x1 1 x2 y1 1 y2
,
M
A(2d, 2c)
N
x
O
P(2a, 0)
b
28. Find the midpoint of each segment to find the endpoints of MN.
OR
AP
29. The endpoints of the midsegment are (
,
) and (
,
).
30. How are the endpoints that your classmate found different from the endpoints that
you found in Exercise 28?
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
__________________________________________________________________________
__________________________________________________________________________
31. What is your classmate’s error? Explain.
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
coordinate geometry
coordinate proof
variable coordinates
Rate how well you can use properties of special figures.
Need to
review
0
2
4
6
8
Now I
get it!
10
177
Lesson 6-8
Name
Class
Date
Additional Problems
6-8
Proofs Using Coordinate Geometry
Problem 1
What are the coordinates of the vertices of the figure below?
ABCD is a rectangle with height b and width 2a. The x-axis
bisects AD and BC, and the y-axis bisects AB and DC.
y
B
A
x
C
D
Problem 2
The diagram shows a general parallelogram with a vertex
at the origin and one side along the x-axis. What are the
coordinates of the point of intersection of the diagonals
of the parallelogram?
y
D (2b, 2c)
C (2a 2b, 2c)
x
A
B (2a, 0)
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83
Name
6-8
Class
Date
Additional Problems (continued)
Proofs Using Coordinate Geometry
Problem 3
Plan a coordinate proof to show
that the midpoints of the sides of
a kite form a rectangle.
a. Name the coordinates of kite
y
K
I
KITE with IE 5 2a along the
x-axis, and KN 5 2b along the
y-axis, and NT 5 2c along the
y-axis.
N
T
b. State the Given and Prove.
c. How will you find the coordinates of the midpoints of
each side?
d. How will you determine whether the figure formed
is a rectangle?
Prentice Hall Geometry • Additional Problems
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84
E
x
Name
6-8
Class
Date
ELL Support
Applying Coordinate Geometry
For Exercises 1–4, match the word in Column A with its definition in Column B.
The first one is done for you.
Column A
Column B
variable
a proof that uses coordinate geometry to show
a theorem is true
1. position
numbers or variables that define the position
of a point on the coordinate plane
2. coordinates
the place where two line segments meet
3. coordinate proof
a symbol or letter that represents an
unknown number
4. vertex
the location of an object
For Exercises 5–7, match the term in Column A with the formula in Column B.
Column A
Column B
5. Distance Formula
( x2 − x1 )2 + ( y2 − y1 )2
6. Slope Formula
! ( x1 + x2 ) ( y1 + y2 ) "
,
#
$
2
2
%
&
( y2 − y1 )
( x2 − x1 )
7. Midpoint Formula
For Exercises 8–10, match the formula in Column A with the problem in
Column B that requires the formula.
Column A
8. Distance Formula
Column B
Find point E halfway between A and B on side AB
of quadrilateral ABCD.
9. Slope Formula
Determine whether sides AB and CD on
quadrilateral ABCD are parallel.
10. Midpoint Formula
Find AB for quadrilateral ABCD on the
coordinate plane.
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71
6-9
Proofs Using
Coordinate Geometry
Vocabulary
Review
1. Circle the Midpoint Formula for a segment in the coordinate plane. Underline the
Distance Formula for a segment in the coordinate plane.
M5 ¢
x1 1 x2 y1 1 y2
,
≤
2
2
d 5 "(x2 2 x1)2 1 (y2 2 y1)2
y 2y
m 5 x2 2 x1
2
1
2. Circle the Midpoint Formula for a segment on a number line. Underline the
Distance Formula for a segment on a number line.
M5
x1 1 x2
2
d 5 |x1 2 x2|
m5
x1 2 x2
2
Vocabulary Builder
VEHR
x and y are
often used as
variables.
ee uh bul
Related Words: vary (verb), variable (adjective)
Definition: A variable is a symbol (usually a letter) that represents one
or more numbers.
Math Usage: A variable represents an unknown number in equations
and inequalities.
Use Your Vocabulary
Underline the correct word to complete each sentence.
3. An interest rate that can change is a variable / vary interest rate.
4. You can variable / vary your appearance by changing your hair color.
5. The amount of daylight variables / varies from summer to winter.
6. Circle the variable(s) in each expression below.
3n
p2 2 2p
41x
4
y
7. Cross out the expressions that do NOT contain a variable.
21m
Chapter 6
9a2 2 4a
36 4 (2 ? 3)
178
8 2 (15 4 3)
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
variable (noun)
Problem 1 Writing a Coordinate Proof
Got It? Reasoning You want to prove that the midpoint of the hypotenuse of a
right triangle is equidistant from the three vertices. What is the advantage of using
coordinates O(0, 0), E(0, 2b), and F(2a, 0) rather than O(0, 0), E(0, b), and F(a, 0)?
8. Label each triangle.
y
E(0,
y
)
, 2b)
E(
M
M
x
O (0, 0)
F(a,
x
O (0, 0)
)
9. Use the Midpoint Formula M 5 ¢
M in each triangle.
x1 1 x2 y1 1 y2
,
≤ to find the coordinates of
2
2
Fisrt Triangle
a
a10
2 ,
, 0)
F(
Second Triangle
1
2
1
a
b 5 a 2,
2
b
a
,
2
0 1 2b
2 b 5(
, b)
10. Use the Distance Formula, d 5 "(x2 2 x1)2 1 (y2 2 y1)2 and your answers to
Exercise 9 to verify that EM 5 FM 5 OM for the first triangle.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
EM
FM
OM
11. Use the Distance Formula, d 5 "(x2 2 x1)2 1 (y2 2 y1)2 and your answers to
Exercise 9 to verify that EM 5 FM 5 OM for the second triangle.
EM
FM
OM
12. Which set of coordinates is easier to use? Explain.
_______________________________________________________________________
_______________________________________________________________________
179
Lesson 6-9
Writing a Coordinate Proof
Problem 2
Got It? Write a coordinate proof of the Triangle
y
Midsegment Theorem (Theorem 5-1).
B(
,
)
Given: E is the midpoint of AB and
F is the midpoint of BC
Prove: EF 6 AC, EF 5 12AC
Use the diagram at the right.
F(
,
E(
)
+
,
)
x
13. Label the coordinates of point C.
O C(
14. Reasoning Why should you make the coordinates
of A and B multiples of 2?
,
A(
)
,
)
b 5(
,
_______________________________________________________________________
_______________________________________________________________________
15. Label the coordinates of A and B in the diagram.
16. Use the Midpoint Formula to find the coordinates of E and F. Label the
coordinates in the diagram.
a
+
coordinates of F
+
,
2
b 5(
2
,
)
17. Use the Slope Formula to determine whether EF 6 AC.
a
+
2
,
+
2
2
slope of EF 5
5
2
2
slope of AC 5
5
2
18. Is EF 6 AC? Explain.
_______________________________________________________________________
_______________________________________________________________________
1
19. Use the Distance Formula to determine whether EF 5 2 AC.
EF 5
AC 5
(
Å
(
Å
20. 12 AC 5 12 ?
Chapter 6
2
)2 1 (
)2 1 (
2
5
5 EF
2
2
)2 5 Î (a)2 1 (0)2 5
)2 5 Î (2a)2 1 (0)2 5
180
)
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
coordinates of E
Lesson Check • Do you know HOW?
Use coordinate geometry to prove that the diagonals of a rectangle are congruent.
21. Draw rectangle PQRS with P at (0, 0).
22. Label Q(a,
), R(
y
, b), and S(
,
).
23. Complete the Given and Prove statements.
Given: PQRS is a
x
Prove: PR >
.
24. Use the Distance Formula to find the length of eatch diagonal.
PR 5
QS 5
25. PR 5
Ä
(
2
)2 1 (
2
)2 5
(
2
)2 1 (
2
)2 5
Ä
, so PR >
.
Lesson Check • Do you UNDERSTAND?
y
Error Analysis Your classmate places a trapezoid on the coordinate
plane. What is the error?
P(b, c) Q(a − b, c)
x
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
26. Check whether the coordinates are for an isosceles trapezoid.
OP 5
QR 5
Ä
Ä
)2 1 (c 2
(b 2
)2 1 (0 2
(a 2
O
)2 5
R(a, 0)
)2 5
27. Does the trapezoid look like an isosceles triangle?
Yes / No
28. Describe your classmate’s error.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
proof
theorem
coordinate plane
coordinate geometry
Rate how well you can prove theorems using coordinate geometry.
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Lesson 6-9
Name
6-9
Class
Date
Additional Problems
Proofs Using Coordinate Geometry
Problem 1
y
Use coordinate geometry to prove that
the diagonals of a rhombus with
vertices R(0, b), H(a, 0), M(0, 2b), and
B(2a, 0) are perpendicular.
R
B
x
H
Given: RHMB is a rhombus with
vertices R(0, b), H(a, 0),
M(0,2b), and B(2a, 0).
M
Prove: RM ' BH
Problem 2
y
Write a coordinate proof to show that
the midpoints of the sides of an
equilateral triangle form another
equilateral triangle.
C
Given: ABC is an equilateral triangle
E
D
with coordinates A(0, 0),
B(4a, 0), and C(2a, 2!3a).
Prove: DEF is an equilateral triangle.
x
A
F
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B
Class
Name
Date
Extra Practice
Chapter 6
Lesson 6-1
Find the sum of the interior angle measures of each polygon.
1. octagon
2. 16-gon
3. 42-gon
Find the missing angle measures.
4.
5.
6.
Find the measure of one interior angle and the measure of one exterior angle in each
regular polygon.
7. nonagon
8. 20-gon
9. 45-gon
Lesson 6-2
Find the values of the variables in each parallelogram.
10.
11.
12.
13.
14.
15.
16. Given: PQRS and QDCA
17. Given:
are parallelograms.
Prove: AP = BS
ABCD
M is the midpoint of CD .
Prove: PM || AD
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Class
Name
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Extra Practice (continued)
Chapter 6
18. In the figure, BD = DF . Find DG and EG .
Lesson 6-3
Based on the markings, decide whether each figure must be a parallelogram.
19.
20.
21.
22.
23.
24.
25. Describe how you can use what you know about
parallelograms to construct a point halfway between a
given pair of parallel lines.
26. Given:
ABCD
BX ⊥ AC , DY ⊥ AC
Prove: BXDY is a
parallelogram.
Lessons 6-4 and 6-5
For each parallelogram, determine the most precise name and find the
measures of the numbered angles.
27.
28.
29.
30.
31.
32.
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Extra Practice (continued)
Chapter 6
33. Use the information in the figure.
34.
Explain how you know that ABCD
is a rectangle.
ABCD is a rhombus. What is
the relationship between ∠1 and ∠2?
Explain.
What value of x makes each figure the given special parallelogram?
35. rhombus
36. rectangle
37. rhombus
Lesson 6-6
Find m∠1 and m∠2.
38.
39.
40.
41.
42.
41.
44. Suppose you manipulate the figure so that ∆PAB,
∆PBC, and ∆PCD are congruent isosceles triangles
with their vertex angles at point P. What kind of
figure is ABCD? Be sure to consider all the
possibilities.
Find EF in each trapezoid.
45.
46.
47.
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Extra Practice (continued)
Chapter 6
Lesson 6-7
Graph the given points. Use slope and the Distance Formula to determine the
most precise name for quadrilateral ABCD.
48. A(3, 5), B(6, 5), C(2, 1), D(1, 3)
49. A(−1, 1), B(3, −1), C(−1, −3), D(−5, −1)
Lesson 6-8
Give coordinates for points D and S without using any new variables.
50. parallelogram
51. rhombus
52. isosceles trapezoid
Lesson 6-9
53. A square has vertices at (2a, 0), (0, 2a), (−2a, 0), and (0, −2a). Use
coordinate geometry to prove that the midpoints of the sides of a
square determine the square.
54. In the figure, ∆PQR is an isosceles triangle. Points M and N
are the midpoints of PQ and PR , respectively. Give a
coordinate proof that the medians of isosceles triangle PQR
2b "
!
intersect at H # 0,
$
3 &
%
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Chapter 6 Test
Form G
Do you know HOW?
Find the values of the variables in each parallelogram.
1.
2.
Find the values of the variable(s) in each figure.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Classify each figure as precisely as possible. Explain your reasoning.
12.
13.
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Chapter 6 Test (continued)
Form G
Give the coordinates for points D and E without using any new variables.
Then find the midpoint of DE .
14.
15.
Do You UNDERSTAND?
Reasoning Determine whether each statement is true or false. If true,
explain your reasoning. If false, provide a counterexample.
16. The diagonals of a rectangle always form four congruent triangles.
17. If the diagonals of a quadrilateral are perpendicular, then the quadrilateral must be
a kite.
18. Reasoning Use coordinate geometry to prove
the following:
Given: ∆ABC with vertices (−j, 0), (0, k), (l, 0), and
midpoints M, N, and O of AB , BC , and AC
Prove: The perimeter of ∆MNO is one-half the
perimeter of ∆ABC.
19. Compare and Contrast Explain in what ways a rectangle is similar to and
different from a kite.
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Chapter 6 Part A Test
Form K
Lessons 6-1 through 6-6
Do you know HOW?
Find the value of each variable.
1.
2.
Algebra Find the value of the variable for which ABCD is a parallelogram.
3.
4.
Classify each quadrilateral as precisely as possible.
5.
6.
7.
8.
9.
10.
Find the value of the variables for each figure.
11.
12.
13.
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Chapter 6 Part A Test (continued)
Form K
Lessons 6-1 through 6-6
Find the value of x for which ABCD is a parallelogram.
14.
15.
Find the values of the variables in each figure.
16.
17.
18.
19. AC = 5z – 7, BD = 2z + 11
Do you UNDERSTAND?
20. How can you classify a rhombus with four congruent angles? Explain.
21. Reasoning Explain why drawing a diagonal on any parallelogram will always
result in two congruent triangles.
22. Reasoning A quadrilateral has perpendicular diagonals. Is this enough to
determine what type of quadrilateral it is? Explain.
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