1. Proving that a Quadrilateral is a Parallelogram
Lesson 7.3
Proving that a Quadrilateral is a Parallelogram

2. Essential Question: How can you prove that a quadrilateral is a parallelogram?

3. To show a quadrilateral is a parallelogram only one of these is needed.
Opposite sides are parallel. (Def)
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other. x
180 - x

4. And one other way…
If one pair of sides is parallel and congruent, then the quadrilateral is a parallelogram.
January 11, 2019

5. Example 1: Is this a parallelogram?
110
70
YES!
70
110
Reason:
Opposite angles congruent
Or, consecutive angles supplementary.
January 11, 2019

6. Example 2: Is this a parallelogram?
10 cm
YES!
7 cm
7 cm
10 cm
Reason:
Opposite sides congruent.
January 11, 2019

7. Example 3: Is this a parallelogram?4 6
YES! 6 4
Reason:
Diagonals bisect each other.
January 11, 2019

8. Example 4: Is this a parallelogram?
NO!
7 cm
7 cm
Reason:
One pair of opp. sides ||, but not .
January 11, 2019

9. Example 5: Is this a parallelogram?
9 cm
YES!
9 cm
Reason:
One pair of opp. sides || & .
January 11, 2019

10. Example 6: Show RSTV is a parallelogram.3 4 1 2
2  3  ST || RV (Alt Int s Converse)
4  1  RS || VT (Alt Int s Converse)
RSTV is a parallelogram since opposite sides are parallel.
January 11, 2019

11. Example 7: Finding the side lengths of a Parallelogram
For what values of x and y is the quadrilateral PQRS a parallelogram? PQ = SR x+ 9 = 2x = x PS = QR y = x + 7 y = = 17

12. Your Turn Finding the angles of a Parallelogram
For what values of x and y is the quadrilateral ABCD a parallelogram? 3x – 32 = 2x x = 32 4y = y y = 87 y = 29

13. Summary: Six ways to prove a quadrilateral is a parallelogram.
opp. sides 
opp. sides || x
180 - x
consec. s supp.
opp. s 
diags. bis. e.o.
1 pair sides || and 
January 11, 2019