5.8Use Properties of Parallelograms Theorem 5.18 If a quadrilateral is a parallelogram, then its opposite sides are congruent. P S R Q If PQRS is a parallelogram,

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


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


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


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


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



2. y

LN oo 23y

23y

By Theorem 5.19, opposite angles are congruent.

12337L

K

M

N32 xoz oy



3. z

o180LK o0o 180123 z

57z

By Theorem 5.20, consecutive angles are supplementary.

12337L

K

M

N32 xoz oy

oo 57z



Theorem 5.21

If a quadrilateral is a parallelogram, then its diagonals _________ each other.

and ____QM P S

RQ

SM

bisect

M

____PM RM


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


Pg. 318, 5.8 #2-34 even

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5.8Use Properties of Parallelograms Theorem 5.18 If a quadrilateral is a parallelogram, then its opposite sides are congruent. P S R Q If PQRS is a parallelogram,