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6-4
6-4
Special Parallelograms
1. Plan
What You’ll Learn
Check Skills You’ll Need
• To use properties of
PACE is a parallelogram and mlPAC ≠ 109.
Complete each of the following.
diagonals of rhombuses and
rectangles
• To determine whether a
parallelogram is a rhombus
or a rectangle
. . . And Why
Lesson 6-2
4.75
4.5
R
6. CP = j 9.5
4
Finding Angle Measures
Finding Diagonal Lengths
Identifying Special
Parallelograms
Real-World Connection
The definitions in Lesson 6-1 of
rhombus, rectangle, and square
included more information than
necessary. Definitions with less
restrictive conditions can often
be used. For example, a rectangle
can be defined as a parallelogram
with one right angle. The fact
that it has four right angles
follows from the properties of a
parallelogram. Using a definition
with less restrictive conditions
makes establishing sufficient
conditions easier.
R
P
Q
Theorem 6-9
Each diagonal of a rhombus bisects two angles of the rhombus.
Proof of Theorem 6-9
More Math Background: p. 304D
Given: rhombus ABCD
B
Prove: AC bisects &BAD and &BCD.
A segment bisects an
angle if and only if it
divides the angle into two
congruent angles.
1
2
3
Math Background
S
Vocabulary Tip
To use properties of diagonals
of rhombuses and rectangles
To determine whether a
parallelogram is a rhombus or
a rectangle
Examples
E
9. Draw a rhombus that is not a square. Draw a rectangle that is not a square.
Explain why each is not a square. See back of book.
If you draw two congruent isosceles triangles
with base PQ, you have drawn a rhombus.
Note that &RPQ, &SPQ, &RQP, and &SQP
are all congruent. This suggests Theorem 6-9
and its proof.
Proof
1
2
C
Diagonals of Rhombuses and Rectangles
Key Concepts
Objectives
3.5
3. m&CEP = j 109 4. PR = j 4.75
5. RE = j 3.5
7
A
2. EP = j 7
7. m&EPA = j 71 8. m&ECA = j 71 P
To lay out a rectangular patio,
as in Example 4
1
1. EC = j 4.5
GO for Help
3
4
C
Lesson Planning and
Resources
1
Proof: ABCD is a rhombus, so its sides are
2
A
D
all congruent. AC > AC by the Reflexive
Property of Congruence. Therefore,
#ABC > #ADC by the SSS Postulate. &1 > &2 and &3 > &4 by CPCTC.
Therefore, AC bisects &BAD and &BCD by the definition of bisect.
See p. 304E for a list of the
resources that support this lesson.
You can show similarly that BD bisects &ABC and &ADC.
PowerPoint
Bell Ringer Practice
The diagonals of a rhombus provide an interesting application of the Converse of
the Perpendicular Bisector Theorem.
Check Skills You’ll Need
For intervention, direct students to:
Properties: Sides and Angles
Lesson 6-4 Special Parallelograms
Special Needs
Below Level
L1
Some students think that because the sides of a
rhombus are congruent its diagonals must also be
congruent. Draw or have students draw several
counterexamples.
learning style: visual
329
L2
Have students use paper folding to explore and
explain Theorems 6-9 and 6-10.
learning style: tactile
Lesson 6-2: Examples 1, 2
Extra Skills, Word Problems, Proof
Practice, Ch. 6
Properties: Diagonals and
Transversals
Lesson 6-2: Example 3
Extra Skills, Word Problems, Proof
Practice, Ch. 6
329
2. Teach
In the rhombus at the right, points B and D are
equidistant from A and C. By the Converse of the
Perpendicular Bisector Theorem, they are on the
perpendicular bisector of AC. This proves the next
theorem. In Exercise 22, you will prove it a second way.
Guided Instruction
Key Concepts
Alternate Method
Challenge students to suggest a
way to prove that each diagonal
of a rhombus bisects two angles
of the rhombus without using
congruent triangles. They could
use the Isosceles Triangle Theorem
and a property of parallel lines to
show that &1 &2 &3 &4.
1
EXAMPLE
3
A
2
A
D
Finding Angle Measures
EXAMPLE
MNPQ is a rhombus and m&N = 120.
Find the measures of the numbered angles.
N
m&1 = m&3 Isosceles Triangle Theorem
Real-World
m&1 + m&3 + 120 = 180
Connection
The diagonals of the rhombus
formed by the pantograph
stay perpendicular when the
pantograph lifts or lowers.
Quick Check
2(m&1) + 120 = 180
2(m&1) = 60
m&1 = 30
1
Triangle Angle-Sum Theorem
P
2
Q
M
Substitute.
3
4
120
Subtract 120 from each side.
Divide each side by 2.
Therefore, m&1 = m&3 = 30. By Theorem 6-9, m&1 = m&2 and m&3 = m&4.
Therefore, m&1 = m&2 = m&3 = m&4 = 30.
50
1
1 Find the measures of the numbered angles in
the rhombus. l1 ≠ 90, l2 ≠ 50, l3 ≠ 50, l4 ≠ 40
C
1
C
You can use Theorems 6-9 and 6-10 to find angle measures in rhombuses.
1 Find the measures of the
numbered angles in the rhombus.
78°
D
B
AC ' BD
pantograph
Additional Examples
B
C
The diagonals of a rhombus are perpendicular.
1
PowerPoint
A
Theorem 6-10
Teaching Tip
In Quick Check, point out that the
diagonals form eight angles at
the vertices of the rhombus.
When the measure of one angle
is known, the other seven angle
measures can be found using
properties of a rhombus.
B
4
2
3
The diagonals of a rectangle, another parallelogram, also have a special property.
4
D
ml1 ≠ 78, ml2 ≠ 90,
ml3 ≠ 12, ml4 ≠ 78
Key Concepts
Theorem 6-11
The diagonals of a rectangle are congruent.
2 One diagonal of a rectangle
has length 8x + 2. The other
diagonal has length 5x + 11.
Find the length of each diagonal.
26, 26
Proof
Proof of Theorem 6-11
Given: Rectangle ABCD
Prove: AC > BD
A
D
B
C
Proof: ABCD is a rectangle, so it is also a parallelogram.
AB > DC because opposite sides of a parallelogram are congruent.
BC > BC by the Reflexive Property of Congruence.
&ABC and &DCB are right angles by the definition of rectangle.
&ABC > &DCB because all right angles are congruent.
#ABC > #DCB by SAS. AC > BD by CPCTC.
330
Chapter 6 Quadrilaterals
Advanced Learners
English Language Learners ELL
L4
Have students rewrite Theorems 6-12, 6-13, and 6-14
so that the parallelogram must be a square.
330
learning style: verbal
Review the terminology of conditionals by examining
Theorems 6-12, 6-13, and 6-14. Point out that each
hypothesis contains two conditions, the first in each
being that the figure is a parallelogram.
learning style: verbal
Guided Instruction
4
5
A
A
D
B
B
E
D
C
C
Finding Diagonal Length
E
D
C
B
EXAMPLE
E
D
C
B
A
3
C
B
A
2
C
B
A
1
2
E
D
E
D
E
Test-Taking Tip
Because FD 5 GE, you
can substitute the
value of y into the
expression for GE to
check your work.
Multiple Choice Find the length of FD in rectangle
GFED if FD = 2y + 4 and GE = 6y - 5.
2 12
3 14
4 12
8 12
F
E
2y + 4 = 6y - 5
G
D
Diagonals of a rectangle are congruent.
9 = 4y
Subtract 2y from each side and
add 5 to each side.
9 =y
4
Divide each side by 4.
Theorems 6-12, 6-13, and 6-14 can
be proved as class exercises when
they are presented. Encourage
students to suggest as many plans
as they can to prove each theorem.
3
1
FD = 2 Q 94 R + 4 = 17
2 , or 8 2. The correct answer is D.
Quick Check
Teaching Tip
2 Find the length of the diagonals of GFED if FD = 5y - 9 and GE = y + 5. 8 1
2
Visual Learners
Students may wish to try drawing
the figures described before
answering the questions.
4
2
1
EXAMPLE
EXAMPLE
Careers
Making sure that an angle is
right or that a line is straight is
necessary for a carpenter building
the frame of a house or a surveyor
marking off a plot of land.
Is the Parallelogram a Rhombus or a Rectangle?
The following theorems are the converses of Theorems 6-9, 6-10, and 6-11,
respectively. You will prove these theorems in the Exercises.
PowerPoint
Key Concepts
Theorem 6-12
Additional Examples
If one diagonal of a parallelogram bisects two angles of the parallelogram, then
the parallelogram is a rhombus.
Theorem 6-13
If the diagonals of a parallelogram are perpendicular, then the parallelogram is
a rhombus.
Theorem 6-14
If the diagonals of a parallelogram are congruent, then the parallelogram is a
rectangle.
4 Explain how you could use
the properties of diagonals to
stake the vertices of a play area
shaped like a rhombus. Sample:
Position ropes at right angles
to each other at their midpoints,
and stake the endpoints.
You can use Theorems 6-12, 6-13, and 6-14 to classify parallelograms.
3
EXAMPLE
3 The diagonals of ABCD are
perpendicular. AB = 16 cm and
BC = 8 cm. Can ABCD be a
rhombus or rectangle? Explain.
No; perpendicular diagonals in a
parallelogram mean that the
figure is a rhombus, but ABCD
is not a rhombus because its
side lengths are not equal.
Identifying Special Parallelograms
Can you conclude that the parallelogram is a rhombus or a rectangle? Explain.
a.
b.
A
D
For: Quadrilateral Activity
Use: Interactive Textbook, 6-4
Quick Check
Yes. A diagonal bisects two
angles. By Theorem 6-12, this
parallelogram is a rhombus.
Resources
• Daily Notetaking Guide 6-4 L3
• Daily Notetaking Guide 6-4—
L1
Adapted Instruction
B
AC BD
C
Yes. The diagonals are congruent.
By Theorem 6-14, this
parallelogram is a rectangle.
3 A parallelogram has angles of 30°, 150°, 30°, and 150°. Can you conclude that it is a
rhombus or a rectangle? Explain. Sample: No; there is not enough information
to conclude that the parallelogram is a rhombus. It cannot be a rectangle
because it has no right angles.
Lesson 6-4 Special Parallelograms
331
Closure
A quadrilateral has congruent
diagonals. Explain why it may or
may not be a rectangle. Sample:
If congruent diagonals bisect
each other, the figure must be
a rectangle by Theorem 6-14.
If they do not bisect each other,
the quadrilateral is not even a
parallelogram, so it is not a
rectangle by the contrapositive
of Theorem 6-3.
331
3. Practice
You can use properties of diagonals to construct special parallelograms.
4
Assignment Guide
Connection
Community Service Builders use properties of
diagonals to “square up” rectangular shapes like
building frames and playing-field boundaries.
1 A B 1-15, 22, 44-46, 50-59
2 A B
16-21, 23-43, 47-49
C Challenge
60-63
Test Prep
Mixed Review
Real-World
EXAMPLE
Suppose you are on the volunteer building team
at the right. You are helping to lay out a rectangular
patio. Explain how to use properties of diagonals
to locate the four corners.
64-66
67-89
To locate the corners, you can use two theorems:
Homework Quick Check
• Theorem 6-7: If the diagonals of a quadrilateral
bisect each other, then the quadrilateral is a parallelogram.
To check students’ understanding
of key skills and concepts, go over
Exercises 6, 20, 23, 38, 46.
Exercises 1–9 Have students state
which property of a rhombus they
use to find each angle measure.
• Theorem 6-14: If the diagonals of a parallelogram are congruent, then the
parallelogram is a rectangle.
Real-World
Connection
A well-planned volunteer
effort can frame a small house
in a day.
Exercises 10–15 Each exercise
Quick Check
requires students to remember
that the diagonals of a rectangle
are congruent. Before having
students begin this exercise set,
make sure that they recall this
property.
First, cut two pieces of rope that will be the diagonals of the foundation rectangle.
Cut them the same length because of Theorem 6-14. Join them at their midpoints
because of Theorem 6-7. Then pull the ropes straight and taut. The ends of the
ropes will be the corners of a rectangle.
4 Kate thinks that they can adapt this method slightly to stake off a square play area.
Is she right? Explain. Yes; if the ropes are # to each other, then the endpoints
of the ropes determine a square.
EXERCISES
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
Practice and Problem Solving
A
Practice by Example
Find the measures of the numbered angles in each rhombus.
1.
Example 1
GO for
Help
GPS Guided Problem Solving
L3
L4
Enrichment
L2
Reteaching
Practice
Name
Class
1
4.
4. 33.5, 33.5, 113, 33.5
5. 32, 90, 58, 32
Special Parallelograms
2.
7.
54ⴗ
1
4
4
2 3
4.
1
4
30ⴗ
4
2
3
2
4
9
60ⴗ
1
2 3
4
68ⴗ
8.
9.
5
(page 331)
3.5
9
9
3.5
9
5
10–15.
See margin.
10. HJ = x and IK = 2x - 7
11. HJ = 3x + 5 and IK = 5x - 9
12. HJ = 3x + 7 and IK = 6x - 11
13. HJ = 19 + 2x and IK = 3x + 22
For each rhombus, (a) find the measures of the numbered angles, and then
(b) find the area.
15. AC = 8 in.
BD = 22 in.
16 cm
29ⴗ
9 cm
1 2
3
20ⴗ
4
D
16.
A
1
2
52ⴗ
B
1
3 4
C
10 m
2
3
13 m
4
© Pearson Education, Inc. All rights reserved.
HIJK is a rectangle. Find the value of x and the length of each diagonal.
14.
2
18. Opposite angles are congruent and supplementary, but the quadrilateral
is not a rectangle.
332
4
3
2
1
58⬚
8.
30⬚
9.
2
3
2
1 2
3
35⬚
90, 55, 90
332
10. LN = x and MP = 2x - 4
11. LN = 5x - 8 and MP = 2x + 1
12. LN = 3x + 1 and MP = 8x - 4
13. LN = 9x - 14 and MP = 7x + 4
14. LN = 7x - 2 and MP = 4x + 3
15. LN = 3x + 5 and MP = 9x - 10
Chapter 6 Quadrilaterals
Determine whether the quadrilateral can be a parallelogram. If not, write
impossible. Explain your answer.
17. One pair of opposite sides is parallel, and the other pair is congruent.
6.
3
1 60⬚
35⬚ 1
60, 90, 30
55, 35, 55, 90
2
Example 2 x Algebra LMNP is a rectangle. Find the value of x and the length of each diagonal.
2
6.
3
The parallelograms below are not drawn to scale. Can the parallelogram
have the conditions marked? If not, write impossible. Explain your answer.
7.
5.
4
3
118, 31, 31
26, 128, 128
106ⴗ 3
3
1
5. 1
1
59ⴗ
2
3
1
118⬚
3
113⬚
4
3.
18ⴗ
3.
26⬚
1
4
1
For each parallelogram, (a) choose the best name, and then (b) find the
measures of the numbered angles.
1.
2
3
1
2
2
104⬚
2
L3
Date
Practice 6-4
2.
4
38, 38,
38, 38
6. 90, 60, 60, 30
L1
Adapted Practice
3
(page 330)
10. 4; LN ≠ MP ≠ 4
13. 9; LN ≠ MP ≠ 67
11. 3; LN ≠ MP ≠ 7
2
14. 35 ; LN ≠ MP ≠ 29
3 ≠ 93
12. 1; LN ≠ MP ≠ 4
15. 52 ; LN ≠ MP ≠ 12 12
Diversity
Is the parallelogram a rhombus or a rectangle? Justify your answer.
Example 3
(page 331)
16.
17.
Exercise 19 Depending on where
students live, older screen doors
may not be familiar sights.
18.
Exercise 21 Ask: What is a name
for a rectangle that is also a
rhombus? square
16–18. See back of book.
19. Hardware You can use a simple device called
a turnbuckle to “square up” structures that are
parallelograms. For the gate pictured at the
right, you tighten or loosen the turnbuckle
on the diagonal cable so that the cable stays
congruent to the other diagonal. Explain why
a frame that normally is rectangular will, when
it sags, keep the shape of a parallelogram.
See margin.
20. Carpentry A carpenter is building a bookcase.
How can she use a tape measure to check
that the bookshelf is rectangular? Justify your
answer and name any theorems used.
See margin.
21. Reasoning Suppose the diagonals of a
parallelogram are both perpendicular and
Turnbuckle
congruent. What type of special quadrilateral
is it? Explain your reasoning. See back of book.
Example 4
(page 332)
B
Apply Your Skills
Exercise 24 Some students may
select answer choice C for square
because it satisfies the condition.
Call attention to the term must
which indicates that the most
general answer possible, choice A
rhombus, is best. Point out that
a square is a rhombus but a
rhombus is not necessarily
a square.
Exercises 35, 36 Suggest that
students add symbols for
trapezoids and kites after
the appropriate properties in
Exercises 25–34 to summarize
all their work in Chapter 6.
22. Developing Proof Complete the flow proof of Theorem 6-10.
22a. Def. of a rhombus
b. Diagonals of a ~
bisect each other.
c. AE O AE
d. Reflexive Prop. of O
e. kABE O kADE
f. CPCTC
g. l Add. Post.
h. lAEB and lAED
are rt. '.
i. O suppl. ' are rt. '
Thm.
j. Def. of #
Given: ABCD is a rhombus.
C
E
A
the methods for solving a system
of equations.
ABCD is a rhombus.
Given
Prove: AC ' BD
B
Exercise 46 If necessary, review
AB AD
a. 9
D
BE DE
b. 9
c. 9
d. 9
e. 9
SSS
m⬔AEB + m⬔AED =180
g. 9
⬔AEB ⬔AED
f. 9
19. The pairs of opp. sides
of the frame remain O,
so the frame remains
a ~.
h. 9
i. 9
20. After measuring the
sides, she can measure
the diagonals.
AC # BD
j. 9
Proof
23. Prove Theorem 6-13.
Given: $ABCD; AC ' BD at E.
GO
nline
Homework Help
Visit: PHSchool.com
Web Code: aue-0604
B
C
E
Prove: ABCD is a rhombus.
A
D
See margin.
24. Multiple Choice A diagonal of a parallelogram bisects one angle of the
parallelogram. What kind of quadrilateral must the figure be? A
rhombus
rectangle
square
cannot tell
Lesson 6-4 Special Parallelograms
333
23. Answers may vary.
Sample: The diagonals
of a ~ bisect each other
so AE O CE. Both
lAED and lCED are
right ' because AC #
BD, and since DE O DE
by the Reflexive Prop.,
kAED O kCED by SAS.
By CPCTC AD O CD,
and since opp. sides of
a ~ are O, AB O BC O
CD O AD. So ABCD is a
rhombus because it has
4 O sides.
333
4. Assess & Reteach
Using Symbols Create your own distinctive symbols for a parallelogram, rhombus,
rectangle, and square. Then copy the properties in Exercises 25–34. After each
property, use your symbols to list the quadrilaterals having that property.
25–34. See
25. All sides are .
26. Opposite sides are .
back of book.
27. Opposite sides are 6.
28. Opposite ' are .
PowerPoint
Lesson Quiz
1. The diagonals of a rectangle
have lengths 4 + 2x and
6x - 20. Find x and the length
of each diagonal.
6; each diagonal has length
16.
1 2
32. Diagonals are .
33. Diagonals are #.
34. Each diagonal bisects opposite '.
37. a quadrilateral that is not a special quadrilateral
38. Writing Summarize the properties of squares that follow from a square being
(a) a parallelogram, (b) a rhombus, and (c) a rectangle. See back of book.
70°
Proof
2
Proof
4
4. Each diagonal is 15 cm long,
and one angle of the
quadrilateral has measure 45.
Impossible;
if diagonals of a parallelogram
are congruent, the
quadrilateral is a rectangle, but
a rectangle has four right
angles.
C
A
Given: $ABCD; AC > BD
B
D
Prove: ABCD is a rectangle.
See back of book.
C
Reasoning Decide whether each of these is a good definition. Justify your answer.
GO for Help
41. A rectangle is a quadrilateral with four right angles. 41–43. See margin.
To review what makes a
good definition, see
Lesson 2-2.
42. A rhombus is a quadrilateral with four congruent sides.
43. A square is a quadrilateral with four right angles and four congruent sides.
x 2 Algebra Find the value(s) of the variable(s) for each parallelogram.
44. RZ = 2x + 5,
SW = 5x - 20
R
5. The diagonals are congruent,
perpendicular and bisect each
other.
Yes; if diagonals of a
parallelogram are congruent,
the quadrilateral is a
rectangle, and if diagonals of a
parallelogram are
perpendicular, the
quadrilateral is a rhombus,
and a rectangle that is a
rhombus is a square.
45. m&1 = 3y - 6
46. BD = 4x - y + 1
GPS
B
W
9x 1
Z
S
2x 1 3y 5
6z
A
T
C
30
x ≠ 5, y ≠ 32,
z ≠ 7.5
D
x ≠ 7.5, y ≠ 3
Open-Ended Given two segments with lengths a and b (a u b), what special
quadrilaterals can you sketch that meet these conditions? Show each sketch.
47–49. See back of book.
47. Both diagonals have length a.
48. The two diagonals have lengths a and b.
49. One diagonal has length a, one side of the quadrilateral has length b.
334
41. Yes; since all right ' are
O, the opp. ' are O and
it is a ~. Since it has all
right ', it is a rectangle.
D
3 4
21
B
Prove: ABCD is a rhombus.
See back of book.
40. Prove Theorem 6-14.
ml1 ≠ 90, ml2 ≠ 20,
ml3 ≠ 20, ml4 ≠ 70
Determine whether the
quadrilateral can be a
parallelogram. If not, write
impossible. Explain.
A
39. Prove Theorem 6-12.
Given: ABCD is a parallelogram;
AC bisects &BAD and &BCD.
1
334
36. a kite
3
ml1 ≠ 62, ml2 ≠ 62,
ml3 ≠ 56
3
31. Diagonals bisect each other.
35. a trapezoid
56°
3.
30. Consecutive ' are supplementary.
Which, if any, of the properties in Exercises 25–34 can the following type of
quadrilateral have? Draw diagrams to illustrate. 35–37. See back of book.
Find the measures of the
numbered angles in each
rhombus.
2.
29. All ' are right '.
Chapter 6 Quadrilaterals
42. Yes; 4 sides are O, so
the opp. sides are O
making it a ~. Since it
has 4 O sides it is also
a rhombus.
43. Yes; a quad. with 4 O
sides is a ~ and a ~
with 4 O sides and 4
right ' is a square.
60. Impossible; if the
diagonals of a ~ are O,
then it would have to be
a rectangle and have
right '.
In Exercises 54–59, the
“given diagonals” are
two segments you
draw. The constructed
figures must have
diagonals that match.
16, 16
50. AC = 2(x - 3) and BD = x + 5
52. AC =
3y
5
and BD = 3y - 4 1, 1
2, 2
51. AC = 2(5a + 1) and BD = 2(a + 1)
1, 1
53. AC = 3c
9 and BD = 4 - c
Constructions Explain how to construct each figure, given its diagonals.
54–59. See back of book.
55. rectangle
56. rhombus
GPS 54. parallelogram
57. square
C
Alternative Assessment
x 2 Algebra ABCD is a rectangle. Find the length of each diagonal.
Problem Solving Hint
Challenge
58. kite
59. trapezoid
Have students draw theorems
without words to illustrate each
of the theorems in Lesson 6-4.
For example, the figure below
illustrates Theorem 6-9: Each
diagonal of a rhombus bisects
two angles of the rhombus.
Determine whether the quadrilateral can be a parallelogram. If not, write
impossible. Explain. 60–62. See margin.
60. The diagonals are congruent, but the quadrilateral has no right angles.
Test Prep
61. Each diagonal is 3 cm long and two opposite sides are 2 cm long.
62. Two opposite angles are right angles, but the quadrilateral is not a rectangle.
Resources
Proof
63. In Theorem 6-12, replace “two angles” with “one angle.” Write a paragraph that
proves this new statement true or show a counterexample to prove it false.
See margin.
Test Prep
Multiple Choice
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 361
• Test-Taking Strategies, p. 356
• Test-Taking Strategies with
Transparencies
64. The diagonals of a quadrilateral are perpendicular bisectors of each other.
What name best describes the quadrilateral? D
A. rectangle
B. parallelogram
C. quadrilateral
D. rhombus
65. The diagonals of a quadrilateral bisect both pairs of opposite angles. What
name best describes the quadrilateral? J
F. parallelogram
G. quadrilateral
H. rectangle
J. rhombus
Short Response
66. Given: QRST is a rhombus, QS intersects RT at P, QR = 9 cm,
and QP = 4.5 cm. Find m&RST. Explain your work. See margin.
Mixed Review
GO for
Help
Lesson 6-3
Can you conclude that the quadrilateral is a parallelogram? Explain.
5
6
67.
68.
69. B
4
Lesson 5-1
4
25
A
73. SU 6 9
74. TU 6 9
75. PQ 6 9
ST
RQ
RP
Lesson 3-4 x 2 76. Algebra Find the value of c. 89
(c 28)
D
R
70. TQ = 9 6 71. PQ = 916 72. TU = 9 5
61. Yes; O diagonals in a ~
mean it can be a
rectangle with two
opposite sides 2 cm
long.
AB ≠ CD and BC ≠ DA.
So AB ≠ BC ≠ CD ≠
DA, and ~ABCD is a
rhombus. The new
statement is true.
25
5
6
67–69. See margin.
In kPQR, points S, T, and U are midpoints.
Complete each statement.
lesson quiz, PHSchool.com, Web Code: aua-0604
C
5
P
S
8
6
U
Q
c
(c 65)
Lesson 6-4 Special Parallelograms
62. Impossible; in a ~,
consecutive ' must be
supp., so all ' must be
right. This would make it
a rectangle.
T
335
63. Given ~ABCD with
diagonal AC. Let AC
bisect lBAD. Because
kABC O kDAC, AB ≠
DA by CPCTC. But since
opp. sides of a ~ are O,
66. [2] Since diagonals of a
rhombus bisect each
other, QS ≠ 9 cm.
Also, since all sides
are O, RS ≠ 9 cm.
So kQRS is an
equilateral k and
each interior l is 180
3
or 608. kQTS is also
an equilateral k, so
its ' are 608. By l add.
(mlPST ± mlPSR ≠
mlRST), mlRST ≠
60 ± 60 or 120.
[1] no work shown OR a
response that is only
partially correct
335