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5.9 Show that a Quad. is a Parallel.
Theorem 5.22
If both pairs of opposite ________ of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
____,BC and ____AB If CD
A D
CB
then ABCD is a parallelogram.
AD
sides
Theorem 5.23
If both pairs of opposite ________ of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
____,B and ____A If CA D
CB
then ABCD is a parallelogram.
D
angles
5.9 Show that a Quad. is a Parallel.
Theorem 5.24
If one pair of opposite sides of a quadrilateral are __________ and ________, then the quadrilateral is a parallelogram.
A D
CB
congruent parallel
,AD __BC and AD __BC If then ABCD is a parallelogram.
5.9 Show that a Quad. is a Parallel.
Theorem 5.25
If the diagonals of a quadrilateral ______ each other, then the quadrilateral is a parallelogram.
A D
CB
bisect
other,each _____ AC and BD If bisectthen ABCD is a parallelogram.
M
5.9 Show that a Quad. is a Parallel.
5.9 Show that a Quad. is a Parallel.
Example 1 Identify parallelogramsIdentify parallelograms
Explain how you know that quadrilateral ABCD is a parallelogram.
a. BA
CD
120
12060
By the _____________________________ you know that
_____,DCBA mmmmCorollary to Theorem 5.16
o360_____.B so m o60
Because both pairs of opposite angles are ___________, congruentthen ABCD is a parallelogram by ______________. Theorem 5.23
o60
5.9 Show that a Quad. is a Parallel.
Example 1 Identify parallelogramsIdentify parallelograms
Explain how you know that quadrilateral ABCD is a parallelogram.
b.
In the diagram AE = ____ and BE = ____. ECSo, the diagonals bisect each other, and ABCD is a parallelogram by ______________. Theorem 5.25
A
D
B
C
x2
x2
E
x4
x4
ED
5.9 Show that a Quad. is a Parallel.Checkpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 1.1. In In quadrilateral GHJK, GHJK,
K. Find .55J and o mm What theorem can,125H ,55G oo mm
you use to show that GHJK is a parallelogram.JH
KG
125 55
55
o603KJHG mmmmoooo 360K5512555 moo 360K235 mo251K m
Theorem 5.23 shows that GHJK is a parallelogram.
5.9 Show that a Quad. is a Parallel.
Example 2 Use algebra with parallelogramsUse algebra with parallelograms
For what value of x is quadrilateral PQRS a parallelogram?By Theorem 5.25, if the diagonals of PQRS ______ each other, then it is a parallelogram. You are given that
.____PT that so Find ____.QT x
bisect
ST RTSet the segment lengths equal. ____PT RTSubstitute for PT and for _____.RT ______5 x 92 x
P R
Q
S
92 xx5T
Subtract ____ from both sides.x2_____ x3 9Divide each side by ____ .3___x 3 .___9__2RT and ____5PT __,When x
Quadrilateral PQRS is a parallelogram when x = ___.3 3 15 3 15
3
5.9 Show that a Quad. is a Parallel.Checkpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 2.2. For what value of For what value of xx is quadrilateral is quadrilateral
DFGH a parallelogram. DFGH a parallelogram.
742 xx72 x
2
7x
D H
F Gx2
74 x
5.3
5.9 Show that a Quad. is a Parallel.
Example 3 Use coordinate geometryUse coordinate geometry
Show that quadrilateral KLMN is a parallelogram?One way is to show that a pair of sides are congruent and parallel. Then apply ________________.
4 ,4L
2 ,2K
2 ,4N
0 ,6M
Theorem 5.24First use the Distance Formula to show that KL and MN are ____________.congruent
_________________KL 224 224 8
_________________MN 246 220 8
MN___ KL ,__ MN KL Because 8
5.9 Show that a Quad. is a Parallel.
Example 3 Use coordinate geometryUse coordinate geometry
Show that quadrilateral KLMN is a parallelogram?
4 ,4L
2 ,2K
2 ,4N
0 ,6M
Then use the slope formula to show that KL ____ MN.
___
KL of slope 24 24
1
___
MN of slope 20 46
1
KL and MN have the same slope, so they are _________.parallelKL and MN are congruent and parallel. So, KLMN is a parallelogram by __________________.Theorem 5.24
5.9 Show that a Quad. is a Parallel.Checkpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 3.3. ExplainExplain another method that can be another method that can be
used to show that quadrilateral used to show that quadrilateral KLMN in Example 3 is a KLMN in Example 3 is a parallelogram.parallelogram.
4 ,4L
2 ,2K
2 ,4N
0 ,6M
Draw the diagonals and find the point of intersection.
Show that the diagonals bisect each other using Distance Formula.Apply Theorem 5.25.
5.9 Show that a Quad. is a Parallel.
Pg. 324, 5.9 #2-22 even
