Theorem 5.22

1
Theorem 5.22
If both pairs of opposite ________ of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
sides
then ABCD is a parallelogram.
2
Theorem 5.23
If both pairs of opposite ________ of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
angles
then ABCD is a parallelogram.
3
Theorem 5.24
If one pair of opposite sides of a quadrilateral
are __________ and ________, then the
quadrilateral is a parallelogram.
congruent
parallel
then ABCD is a parallelogram.
4
Theorem 5.25
If the diagonals of a quadrilateral ______ each
other, then the quadrilateral is a parallelogram.
bisect
then ABCD is a parallelogram.
5
Identify parallelograms
Example 1
Explain how you know that quadrilateral ABCD is a
parallelogram.
a.
Corollary to Theorem 5.16
By the _____________________________ you know
that
congruent
Because both pairs of opposite angles are
___________,
Theorem 5.23
then ABCD is a parallelogram by ______________.
6
Identify parallelograms
Example 1
Explain how you know that quadrilateral ABCD is a
parallelogram.
b.
EC
ED
In the diagram AE ____ and BE ____.
So, the diagonals bisect each other,
and ABCD is a parallelogram by ______________.
Theorem 5.25
7
Checkpoint. Complete the following exercises.
Theorem 5.23 shows that GHJK is a parallelogram.
8
Use algebra with parallelograms
Example 2
For what value of x is quadrilateral PQRS a
parallelogram?
bisect
Set the segment lengths equal.
Substitute for PT and for _____.
Subtract ____ from both sides.
Divide each side by ____ .
Quadrilateral PQRS is a parallelogram when x
___.
9
Checkpoint. Complete the following exercises.

1. For what value of x is quadrilateral DFGH a
   parallelogram.

10
Use coordinate geometry
Example 3
Show that quadrilateral KLMN is a parallelogram?
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
show that KL and MN are ____________.
congruent
11
Use coordinate geometry
Example 3
Show that quadrilateral KLMN is a parallelogram?
Then use the slope formula to show that KL ____
MN.
parallel
KL and MN have the same slope, so they are
_________.
KL and MN are congruent and parallel. So, KLMN is
a parallelogram by __________________.
Theorem 5.24
12
Checkpoint. Complete the following exercises.

1. Explain another method that can be used to show
   that quadrilateral KLMN in Example 3 is a
   parallelogram.

Draw the diagonals and find the point of
intersection.
Show that the diagonals bisect each other using
Distance Formula.
Apply Theorem 5.25.
13
Pg. 324, 5.9 2-22 even