6.3 Showing Quadrilaterals are Parallelograms

Goal
Show that a quadrilateral
is a parallelogram.
Key Words
• parallelogram p. 310
Student Help
STUDY TIP
The theorems in this
lesson are the
converses of the
theorems in Lesson 6.2.
316 Chapter 6 Quadrilaterals
6.3 Showing Quadrilaterals are
Parallelograms
Geo-Activity
Making Parallelograms
●1 Cut two straws to form two congruent pairs.
●2 Partly unbend two paper clips, link their smaller ends, and insert the
larger ends into two cut straws, as shown. Join the rest of the straws
to form a quadrilateral with opposite sides congruent, as shown.
●3 Change the angles of your quadrilateral. Is your quadrilateral always
a parallelogram?
The Geo-Activity above describes one way to show that a quadrilateral
is a parallelogram.
THEOREMS 6.6 and 6.7
Theorem 6.6
Words If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Symbols If PQ&* c SR&* and QR&* c PS&*, then
PQRS is a parallelogram.
Theorem 6.7
Words If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Symbols If aP caR and aQ caS, then
PQRS is a parallelogram.
P
P
P
P
S
S
R
R

IStudent Help
I CLASSZONE.COM
MORE EXAMPLES
More examples at
classzone.com
EXAMPLE 1 Use Opposite Sides
Tell whether the quadrilateral is a
parallelogram. Explain your reasoning.
Solution
The quadrilateral is not a parallelogram. It has two pairs
of congruent sides, but opposite sides are not congruent.
EXAMPLE 2 Use Opposite Angles
Tell whether the quadrilateral is a
parallelogram. Explain your reasoning.
140
Solution
The quadrilateral is a parallelogram because both pairs of
opposite angles are congruent.
Use Opposite Sides and Opposite Angles
In Exercises 1 and 2, tell whether the quadrilateral is a parallelogram.
Explain your reasoning.
1. D 5 C
2.
4
A
5
4
B
3. In quadrilateral WXYZ, WX 15, YZ 20, XY 15, and ZW 20.
Is WXYZ a parallelogram? Explain your reasoning.
THEOREM 6.8
Words If an angle of a quadrilateral is
supplementary to both of its
consecutive angles, then the
quadrilateral is a parallelogram.
Symbols If maP maQ 180 and maQ maR 180,
then PQRS is a parallelogram.
L
K
6.3 Showing Quadrilaterals are Parallelograms 317
40
P
y
140
P
x
x y 180
S
x
40
M
J
R

318 Chapter 6 Quadrilaterals
EXAMPLE 3 Use Consecutive Angles
Tell whether the quadrilateral is a parallelogram. Explain your
reasoning.
a. U
V b. E F c.
95
85
125
T
85
Solution
W
120 55
H G
B C
A D
a. aU is supplementary to aT and aV (8595180). So, by
Theorem 6.8, TUVW is a parallelogram.
b. aG is supplementary to aF (55125180), but aG is not
supplementary to aH (55120180). So, EFGH is not a
parallelogram.
c. aD is supplementary to aC (9090180), but you are not
given any information about aA or aB. Therefore, you cannot
conclude that ABCD is a parallelogram.
THEOREM 6.9
Words If the diagonals of a quadrilateral
bisect each other, then the
quadrilateral is a parallelogram.
Symbols If QM&* cMS &*** and PM &** c MR &**,
then PQRS is a parallelogram.
EXAMPLE 4 Use Diagonals
Tell whether the quadrilateral is a parallelogram. Explain your
reasoning.
a. b.
J K
19
18 19
M L
Solution
18
a. The diagonals of JKLM bisect each other. So, by Theorem 6.9,
JKLM is a parallelogram.
b. The diagonals of PQRS do not bisect each other. So, PQRS is
not a parallelogram.
S
P
P
4
2
P
3
3
R
M
P
S
R

Use Consecutive Angles and Diagonals
Tell whether the quadrilateral is a parallelogram. Explain your
reasoning.
4. G 5.
80
H
130
E
60
F
L M
104
76 104
K N
6. R P
7. U V
T
5
4
X
5
4
S
P
T
W
You have learned five ways to show that a quadrilateral is a
parallelogram.
SUMMARY
SHOWING A QUADRILATERAL IS A PARALLELOGRAM
Definition of parallelogram, p. 310
Show that both pairs of opposite sides
are parallel.
Theorem 6.6, p. 316
Show that both pairs of opposite sides
are congruent.
Theorem 6.7, p. 316
Show that both pairs of opposite angles
are congruent.
Theorem 6.8, p. 317
Show that one angle is supplementary
to both of its consecutive angles.
Theorem 6.9, p. 318
Show that the diagonals bisect
each other.
y x
6.3 Showing Quadrilaterals are Parallelograms 319
x
x y 180

6.3
Exercises
Guided Practice
Vocabulary Check
Practice and Applications
Extra Practice
See p. 685.
Skill Check
Homework Help
Example 1: Exs. 8–10
Example 2: Exs. 11–13
Example 3: Exs. 14–16
Example 4: Exs. 17–19
320 Chapter 6 Quadrilaterals
In Exercises 1–3, name all the sides or angles of gEFGH that match
the description.
1. Opposite sides are parallel.
2. Opposite angles are congruent.
3. Consecutive angles are supplementary.
4. Explain why every parallelogram is a quadrilateral, but not every
quadrilateral is a parallelogram.
Decide whether you are given enough information to show that the
quadrilateral is a parallelogram. Explain your reasoning.
5.
50
6. 7.
6
6
Using Opposite Sides Tell whether the quadrilateral is a
parallelogram. Explain your reasoning.
8. 9. 10.
8 6
Using Opposite Angles Tell whether the quadrilateral is a
parallelogram. Explain your reasoning.
11. 12. 13.
82 98
98
130
Using Consecutive Angles Tell whether the quadrilateral is a
parallelogram. Explain your reasoning.
14.
60 120
15. 16.
120
82
6
118
72
8
118
H
E
80
3
3
G
110
2
2
F
110 60

Bicycles
DERAILLEURS (named from
the French word meaning
“to derail”) move the chain
to change gears.
Application Links
CLASSZONE.COM
Using Diagonals Tell whether the quadrilateral is a parallelogram.
Explain your reasoning.
17. 18. 19.
20. Bicycle Gears When you change
gears on a bicycle, the derailleur
moves the chain to the new gear.
For the derailleur at the right,
AB 1.8 cm, BC 3.6 cm,
CD 1.8 cm, and DA 3.6 cm.
Explain why AB&* and CD &** are always
parallel when the derailleur moves.
21. Error Analysis What is wrong with the student’s argument below?
A quadrilateral that has one pair
of sides congruent and the other
pair of sides parallel is always a
parallelogram.
You be the Judge
22. Three of the interior angles of a
quadrilateral have measures of 75, 75, and 105. Is this enough
information to conclude that the quadrilateral is a parallelogram?
Explain your answer.
23. Visualize It! Explain why the following method of drawing a
parallelogram works. State a theorem to support your answer.
●1 Use a ruler to draw ●2 Draw another ●3
a segment and its
midpoint.
segment so the
midpoints coincide.
6 4
7
5
B C
A D
Connect the
endpoints of the
segments.
24. Challenge If one pair of opposite sides of a quadrilateral is both
congruent and parallel, is the quadrilateral a parallelogram?
Explain your reasoning.
6.3 Showing Quadrilaterals are Parallelograms 321

Student Help
SKILLS REVIEW
To review the formula
for finding slope, see
p. 665.
Standardized Test
Practice
322 Chapter 6 Quadrilaterals
EXAMPLE Coordinate Geometry
Use the slopes of the segments in
the diagram to determine if the
quadrilateral is a parallelogram.
Solution
Lines and line segments are parallel
if they have the same slope.
Slope of AB&*: 4 0
2 1
4
4
1
0
Slope of DC &*: 4
6 5
4
A(1, 0) 4 D(5, 0) x
4
1
Slope of BC&*: 4 4
6 2
0
0
0 Slope of AD&*: 0
4 5 1
0
0
4
ANSWER The slopes of AB&* and DC &* are the same, so AB&* DC &*. The
slopes of BC&*and AD&* are the same, so AD&* BC&*. Both pairs
of opposite sides are parallel, so ABCD is a parallelogram.
1
y
B(2, 4) C(6, 4)
Coordinate Geometry Use the slopes of the segments in the
diagram to determine if the quadrilateral is a parallelogram.
25. y
26.
1
G(1, 4) H(5, 4)
F(2, 1)
1
J(6, 1)
27. Multi-Step Problem Suppose you shoot a pool ball as shown
below and it rolls back to where it started. The ball bounces off
each wall at the same angle at which it hits the wall.
a. The ball hits the first wall at an angle of 63.
So maAEF maBEH 63. Explain why maAFE 27.
b. Explain why maFGD 63.
c. What is maGHC? maEHB?
x
d. Find the measure of each interior angle of EFGH. What kind
of quadrilateral is EFGH? How do you know?
1
y
P(0, 2)
1
Œ(2, 5) R(5, 5)
S(4, 2)
x

Quiz 1
Mixed Review
Algebra Skills
Finding Angle Measures Find the measure of aA. (Lesson 6.1)
28.
D
99
A 29.
A
B
70
30.
D
110
A
C
85 101
B
114
D
C
C
82 78
B
Finding Measures Find the measure in gJKLM. (Lesson 6.2)
31. Find maK. 32. Find maJ.
33. Find ML. 34. Find KL.
Evaluating Expressions Evaluate the expression for the given value
of the variable. (Skills Review, p. 670)
35. 2x 7 when x 5 36. 4y 3 when y 2
37. 13 3m when m 3 38. 1 b when b 10
39. 5 2a when a 6 40. 8c 5 when c 4
41. 12x x 2 when x 1 42. 3
4 q2 2 when q 4
43. 5n 3 4n when n 2 44. 15p p 2 when p 3
Decide whether the figure is a polygon. If so, tell what type. If not,
explain why. (Lesson 6.1)
1. 2. 3.
Find the values of the variables in the parallelogram. (Lesson 6.2)
4. y
5. x y
6.
16
15
x
Tell whether the quadrilateral is a parallelogram. Explain your
reasoning. (Lesson 6.3)
7.
65
8. 9.
130 50
115 65
122 z
K
6.3 Showing Quadrilaterals are Parallelograms 323
L
14
x
M
71
11
J
y
12
9