1. 1. What congruence postulate shows that ABE CDE ?=
ANSWER
SAS
2. If E G, find m E. =
ANSWER
118º

2. Use properties of parallelograms.
Target
Use properties of parallelograms.
You will…
Use properties to identify parallelograms.

3. Vocabulary
Proving Quadrilaterals are Parallelograms
A quadrilateral is a parallelogram if …
both pairs of opposite sides are parallel – Definition
both pairs of opposite sides are congruent – Theorem 8.7
both pairs of opposite angles are congruent – Theorem 8.8
one pair of opposite sides is congruent and parallel
– Theorem 8.9
its diagonals bisect each other – Theorem 8.10.

4. EXAMPLE 1
Solve a real-world problem
Ride
An amusement park ride has a moving platform attached to four swinging arms. The platform swings back and forth, higher and higher, until it goes over the top and around in a circular motion. In the diagram below, AD and BC represent two of the swinging arms, and DC is parallel to the ground (line l). Explain why the moving platform AB is always parallel to the ground.

5. EXAMPLE 1
Solve a real-world problem
SOLUTION
The shape of quadrilateral ABCD changes as the moving platform swings around, but its side lengths do not change. Both pairs of opposite sides are congruent, so ABCD is a parallelogram by Theorem 8.7.
By the definition of a parallelogram, AB DC . Because DC is parallel to line l, AB is also parallel to line l by the Transitive Property of Parallel Lines. So, the moving platform is parallel to the ground.

6. GUIDED PRACTICE
for Example 1
In quadrilateral WXYZ, m W = 42°, m X =138°, m Y = 42°. Find m Z. Is WXYZ a parallelogram? Explain your reasoning. 1.
138°; yes; the sum of the measures of the interior angles in a quadrilateral is 360°, so the measure of is 138°. Since opposite angles of the quadrilateral are congruent, WXYZ is a parallelogram. Z
ANSWER

7. EXAMPLE 2
Identify a parallelogram
ARCHITECTURE
The doorway shown is part of a building in England. Over time, the building has leaned sideways. Explain how you know that SV = TU.
SOLUTION
In the photograph, ST UV and ST UV. By Theorem 8.9, quadrilateral STUV is a parallelogram.
By Theorem 8.3, you know that opposite sides of a parallelogram are congruent. So, SV = TU.

8. Use algebra with parallelograms
EXAMPLE 3
Use algebra with parallelograms
ALGEBRA
For what value of x is quadrilateral CDEF a parallelogram?
SOLUTION
By Theorem 8.10, if the diagonals of CDEF bisect each other, then it is a parallelogram. You are given that CN EN . Find x so that FN DN .
When x = 4,
FN = DN
Set the segment lengths equal.
FN = 5(4) –8
5x – 8 = 3x
Substitute 5x –8 for FN and 3x for DN.
= 12
2x – 8 =
Subtract 3x from each side.
DN =
3(4)
2x = 8
Add 8 to each side.
= 12
x = 4
Divide each side by 2.

9. GUIDED PRACTICE
for Examples 2 and 3
What theorem can you use to show that the quadrilateral is a parallelogram? 2.
Theorem 8.9
ANSWER
If one pair of opposite sides is parallel and congruent, then the quadrilateral is a parallelogram.

10. GUIDED PRACTICE
for Examples 2 and 3
What theorem can you use to show that the quadrilateral is a parallelogram? 3.
Theorem 8.7
ANSWER
If both pair of opposite sides are congruent, then the quadrilateral is a parallelogram.

11. GUIDED PRACTICE
for Examples 2 and 3
What theorem can you use to show that the quadrilateral is a parallelogram? 4.
Theorem 8.8
ANSWER
If both pair of opposite angles are congruent, then the quadrilateral is a parallelogram.

12. GUIDED PRACTICE
for Examples 2 and 3
For what value of x is quadrilateral MNPQ a parallelogram? Explain your reasoning. 5.
ANSWER
2; The diagonals of a parallelogram bisect
each other so solve 2x = 10 – 3x for x.

13. EXAMPLE 4
Use coordinate geometry
Show that quadrilateral ABCD is a parallelogram.
SOLUTION
One way is to show that a pair of sides are congruent and parallel. Then apply Theorem 8.9.
First use the Distance Formula to show that AB and CD are congruent.
AB = =
[2 – (–3)]2 + (5 – 3)2 29
So, AB
CD.
CD =
(5 – 0)2 + (2 – 0)2 = 29

14. EXAMPLE 4
Use coordinate geometry
Then use the slope formula to show that AB
CD.
5 – (3)
2 – (–3) = 2 5
Slope of AB =
2 – 0
5 – 0 = 2 5
Slope of CD =
Because AB and CD have the same slope, they are parallel.
AB and CD are congruent and parallel.
So, ABCD is a parallelogram by Theorem 8.9.
ANSWER

15. EXAMPLE 4
GUIDED PRACTICE
for Example 4
6. Explain how other methods can be used to show that quadrilateral ABCD in Example 4 is a parallelogram.
ANSWER
Find the slopes of all four sides and show that opposite sides are parallel.
A second way is to find the lengths of each side and show that opposite sides are congruent.
A third way is to find the point of intersection of the diagonals and show the diagonals bisect each other.