5.8 Use Properties of Parallelograms Theorem 5.18 If a quadrilateral is a parallelogram, then its opposite sides are congruent. . ____ QR and RS ____ PQ P S R Q If PQRS is a parallelogram, then PS
Dec 23, 2015
5.8 Use Properties of Parallelograms
Theorem 5.18
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
.____QR and RS____ PQ
P S
RQ
If PQRS is a parallelogram, then
PS
5.8 Use Properties of Parallelograms
Theorem 5.19
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
.S____ and ____P R
P S
RQ
If PQRS is a parallelogram, then
Q
5.8 Use Properties of Parallelograms
Example 1 Use properties of parallelogramsUse properties of parallelograms
Find the value of x and y. F
J H
G
oy
686x
13
SolutionFGHJ is a parallelogram by the definition of a parallelogram.Use Theorem 5.18 to find the value of x.
Opposite sides of a are .Substitute x + 6 for FG and __ for ___.
____FG HJ13 HJ ____6 x 13
Subtract 6 from both sides. ___x 7By Theorem 5.19, ._____For ____,F mH Hm
. ___ so, o y o68In FGHJ, x = ___ and y = _____.7 68
5.8 Use Properties of Parallelograms
Theorem 5.20
If a quadrilateral is a parallelogram, then its consecutive angles are _______________.
. ____oo yx
P S
RQ
If PQRS is a parallelogram, theno180
supplementary
oxoy
ox oy
5.8 Use Properties of Parallelograms
Example 2 Use properties of a parallelogramUse properties of a parallelogram
Gates As shown, a gate contains several parallelograms.
Solution
_____.________ADC m
____DABADC So, mm o180
o65 DAB when ADC Find mm
By Theorem 5.20, the consecutive angle pairs in ABCD are ______________. supplementary
A
D
C
B
,65DAB Because omo180 o65 o115
Checkpoint. Find the indicated Checkpoint. Find the indicated measure in LMN shown at measure in LMN shown at the right.the right.
1. x
LMKN 3732 x
x2 4020x
By Theorem 5.18, opposite sides are congruent.
12337L
K
M
N32 xoz oy
5.8 Use Properties of Parallelograms
Checkpoint. Find the indicated Checkpoint. Find the indicated measure in LMN shown at measure in LMN shown at the right.the right.
2. y
LN oo 23y
23y
By Theorem 5.19, opposite angles are congruent.
12337L
K
M
N32 xoz oy
5.8 Use Properties of Parallelograms
Checkpoint. Find the indicated Checkpoint. Find the indicated measure in LMN shown at measure in LMN shown at the right.the right.
3. z
o180LK o0o 180123 z
57z
By Theorem 5.20, consecutive angles are supplementary.
12337L
K
M
N32 xoz oy
oo 57z
5.8 Use Properties of Parallelograms
5.8 Use Properties of Parallelograms
Theorem 5.21
If a quadrilateral is a parallelogram, then its diagonals _________ each other.
and ____QM P S
RQ
SM
bisect
M
____PM RM
5.8 Use Properties of Parallelograms
Example 3 Find the intersection of diagonalsFind the intersection of diagonals
The diagonals of STUV intersect at point W. Find the coordinates of W.
T
S
W
V
U
Solutionbisect
midpointBy Theorem 5.21, the diagonals of a parallelogram _______ each other. So, W is the _________ of the diagonals TV and SU.Use the _____________________.Midpoint FormulaCoordinates of the midpoint W of
._____________SU ,2
06 2
05,3 2
5
5.8 Use Properties of ParallelogramsCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 4.4. The diagonals of VWXY The diagonals of VWXY
intersect at point Z. Find intersect at point Z. Find the coordinates of Zthe coordinates of Z
W
V
Z
Y
X
By Theorem 5.21, the diagonals of a parallelogram bisect each other. So, Z is the midpoint of the diagonals WY and VX.
2
,
2
VX ofMidpoint 07 06
,
2
73
5.8 Use Properties of ParallelogramsCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 5.5. Given that FGHJ is a Given that FGHJ is a
parallelogram, find MH parallelogram, find MH and FH.and FH.
F
M
J H
G5
By Theorem 5.21, the diagonals of a parallelogram bisect each other. So, M is the midpoint of the diagonal FH.
FMMH 5MH
MHFMFH 55FH 10
5.8 Use Properties of Parallelograms
Pg. 318, 5.8 #2-34 even