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5.9
Show that a Quad. is a Parallel.
Theorem 5.22
sides
If both pairs of opposite ________
of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
B
A
C
D
If AB  ____
CD and BC  ____,
AD
then ABCD is a parallelogram.
5.9
Show that a Quad. is a Parallel.
Theorem 5.23
angles of a
If both pairs of opposite ________
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
B
A
C
D
C and B  ____,
If A  ____
D
then ABCD is a parallelogram.
5.9
Show that a Quad. is a Parallel.
Theorem 5.24
If one pair of opposite sides of a quadrilateral are
congruent and ________,
parallel then the
__________
quadrilateral is a parallelogram.
B
A
C
D
 AD and BC __ AD,
If BC __
then ABCD is a parallelogram.
5.9
Show that a Quad. is a Parallel.
Theorem 5.25
If the diagonals of a quadrilateral ______
bisect each
other, then the quadrilateral is a
parallelogram.
B
C
M
A
D
If BD and AC _____
bisect each other,
then ABCD is a parallelogram.
5.9
Show that a Quad. is a Parallel.
Example 1 Identify parallelograms
Explain how you know that quadrilateral ABCD is a
parallelogram.
a.
D
A
60
120
60
o
B
120
C
By the _____________________________
Corollary to Theorem 5.16 you know that
360
mA  mB  mC  mD  _____,
o
so mB  _____.
60
o
congruent
Because both pairs of opposite angles are ___________,
then ABCD is a parallelogram by ______________.
Theorem 5.23
5.9
Show that a Quad. is a Parallel.
Example 1 Identify parallelograms
Explain how you know that quadrilateral ABCD is a
parallelogram.
b.
A
2x E 4x
4x
2x
D
B
C
EC and BE = ____.
In the diagram AE = ____
ED
So, the diagonals bisect each other,
and ABCD is a parallelogram by ______________.
Theorem 5.25
5.9
Show that a Quad. is a Parallel.
Checkpoint. Complete the following exercises.
1. In quadrilateral GHJK, mG  55o , mH  125o ,
and mJ  55 . Find mK. What theorem can
o
you use to show that GHJK is a parallelogram.
H
G
55
125
55
J
Theorem 5.23
shows that GHJK
is a parallelogram.
K
mG  mH  mJ  mK  360
o
o
o
o
55  125  55  mK  360
o
o
235  mK  360
o
mK  125
o
5.9
Show that a Quad. is a Parallel.
Example 2 Use algebra with parallelograms
For what value of x is quadrilateral
PQRS a parallelogram?
By Theorem 5.25, if the diagonals of
PQRS ______
bisect each other, then it is a
parallelogram. You are given that
QT  ____.
RT .
ST Find x so that PT  ____
PT  ____
RT
5 x  ______
2x  9
__
9
3 x  ___
x  ___
3
P
Q
2x  9
5x
T
R
S
Set the segment lengths equal.
Substitute for PT and for _____.
RT
2 x from both sides.
Subtract ____
Divide each side by ____
3 .
When x  __,
__ and RT  2__
___ .
3   15
3   9  15
3 PT  5__
3
Quadrilateral PQRS is a parallelogram when x = ___.
5.9
Show that a Quad. is a Parallel.
Checkpoint. Complete the following exercises.
2. For what value of x is quadrilateral
DFGH a parallelogram.
2x  4x  7
 2x  7
7
x   3.5
2
F
G
2x
4x  7
D
H
5.9
Show that a Quad. is a Parallel.
Example 3 Use coordinate geometry
L4, 4
Show that quadrilateral
KLMN is a parallelogram?
K2, 2
One way is to show that a pair of
sides are congruent and parallel.
Then apply ________________.
Theorem 5.24
First use the Distance Formula to
congruent
show that KL and MN are ____________.
M6, 0
N4,  2
4  2  4 ____
8
KL  __________
2  ___
2
2
6  4  0  ____
8
MN  __________
 2  ___
2
2
 MN
Because KL  MN  __
8 , KL ___
5.9
Show that a Quad. is a Parallel.
Example 3 Use coordinate geometry
Show that quadrilateral
KLMN is a parallelogram?
L4, 4
K2, 2
Then use the slope formula to
show that KL ____ MN.
42
1
slope of KL 
 ___
42
0   2
1
slope of MN 
 ___
64
M6, 0
N4,  2
parallel
KL and MN have the same slope, so they are _________.
KL and MN are congruent and parallel. So, KLMN is a
Theorem 5.24
parallelogram by __________________.
5.9
Show that a Quad. is a Parallel.
Checkpoint. Complete the following exercises.
3. Explain another method that can be
used to show that quadrilateral
KLMN in Example 3 is a
parallelogram.
L4, 4
K2, 2
M6, 0
N4,  2
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