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5.8
Use Properties of Parallelograms
Theorem 5.18
If a quadrilateral is a parallelogram, then its
opposite sides are congruent.
Q
P
R
S
If PQRS is a parallelogram, then
PQ  RS and QR  ____
____
PS .
5.8
Use Properties of Parallelograms
Theorem 5.19
If a quadrilateral is a parallelogram, then its
opposite angles are congruent.
Q
P
R
S
If PQRS is a parallelogram, then
Q  S.
P  ____
R and ____
5.8
Use Properties of Parallelograms
Example 1 Use properties of parallelograms
x6
Find the value of x and y.
F
68
Solution
FGHJ is a parallelogram by the
definition of a parallelogram.
J
Use Theorem 5.18 to find the value of x.
FG  ____
HJ
x  6  ____
13
x  ___
7
G
yo
13

H
Opposite sides of a
are .
Substitute x + 6 for FG and 13
__ for ___.
HJ
Subtract 6 from both sides.
By Theorem 5.19, F  ____,
_____
H .
H or mF  m
so, y  ___
68 .
o
In
o
FGHJ, x = ___
7 and y = _____.
68
5.8
Use Properties of Parallelograms
Theorem 5.20
If a quadrilateral is a parallelogram, then its
supplementary
consecutive angles are _______________.
Q
P
x
o
y
yo
x
o R
o
S
If PQRS is a parallelogram, then
x  y  ____
180 .
o
o
o
5.8
Use Properties of Parallelograms
Example 2 Use properties of a parallelogram
Gates As shown, a gate contains
several parallelograms.
Find mADC when mDAB  65o
A
D
C
Solution
By Theorem 5.20, the consecutive angle pairs
supplementary
in ABCD are ______________.
So, mADC  mDAB  180
____
o
Because mDAB  65 ,
o
mADC  180
____  ____
65  _____.
115
o
o
o
B
5.8
Use Properties of Parallelograms
Checkpoint. Find the indicated
measure in LMN shown at
the right.
zo
K
1. x
L
123
2x  3
By Theorem 5.18, opposite sides are congruent.
KN  LM
2 x  3  37
2 x  40
x  20
37
M
yo
N
5.8
Use Properties of Parallelograms
Checkpoint. Find the indicated
measure in LMN shown at
the right.
zo
K
2. y
L
123
37
2x  3
By Theorem 5.19, opposite angles are congruent.
N  L
y  23
y  23
o
o
M
yo
N
5.8
Use Properties of Parallelograms
Checkpoint. Find the indicated
measure in LMN shown at
the right.
zo
K
3. z
L
123
2x  3
37
M
yo
N
By Theorem 5.20, consecutive angles are supplementary.
K  L  180
o
z  123  180
o
o
z  57
z  57
o
0
o
5.8
Use Properties of Parallelograms
Theorem 5.21
If a quadrilateral is a parallelogram, then its
bisect
diagonals _________
each other.
Q
R
M
P
S
QM  ____
SM and
PM  ____
RM
5.8
Use Properties of Parallelograms
Example 3 Find the intersection of diagonals
The diagonals of STUV
intersect at point W. Find
the coordinates of W.
Solution
U
T
W
V
S
By Theorem 5.21, the diagonals of a parallelogram _______
bisect
each other. So, W is the _________
midpoint of the diagonals TV and SU.
Use the _____________________.
Midpoint Formula
Coordinates of the midpoint W of
SU  
60 50
,
_________
2
2
 
5
____
3, 2
.
5.8
Use Properties of Parallelograms
Checkpoint. Complete the following exercises.
4. The diagonals of VWXY
intersect at point Z. Find
the coordinates of Z
X
W
Z
V
Y
By Theorem 5.21, the diagonals of a parallelogram bisect
each other. So, Z is the midpoint of the diagonals WY and VX.
 70 60
Midpoint of VX  
,
2
 2
  7 , 
  3
 2 
5.8
Use Properties of Parallelograms
Checkpoint. Complete the following exercises.
5. Given that
FGHJ is a
parallelogram, find MH
and FH.
G
F
5
M
J
By Theorem 5.21, the diagonals of a parallelogram bisect
each other. So, M is the midpoint of the diagonal FH.
MH  FM
MH  5
FH  FM  MH
FH  5  5  10
H
5.8
Use Properties of Parallelograms
Pg. 318, 5.8 #2-34 even