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

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

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

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

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

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

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

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

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

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

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 -1 10 = x PS = QR y = x + 7 y = 10 + 7 = 17

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 + 87 3y = 87 y = 29

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