6-3 Proving a Quad is a parallelogram

Theorem Theorem 6-5: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

3𝑥=𝑥+5 −𝑥 −𝑥 2𝑥=5 2 2 𝑥= 5 2 =2.5 2𝑦−7=𝑦+2 −𝑦 −𝑦 𝑦−7=2 + 7 +7 𝑦=9

Theorem 6-6: If one pair of opposite sides is both congruent and parallel, then the quadrilateral is a parallelogram.

𝑎+40+𝑎=180 2𝑎+40=180 −40 −40 2𝑎=140 2 2 𝑎=70 3𝑐−3=𝑐+1 −𝑐 −𝑐 2𝑐−3=1 +3 +3 2𝑐=4 2 2 𝑐=2

Theorems Theorem 6-7: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6-8: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

5 ways to prove that a quadrilateral is a parallelogram: Both pairs of opposite sides are parallel (Definition) Diagonals bisect each other (Thm 6-5) One pair of opposite sides are both congruent and parallel (Thm. 6-6) Both pairs of opposite sides are congruent (Thm. 6-7) Both pairs of opposite angles are congruent (Thm. 6-8)

1. No, 2. Yes, 3. No

Practice Pg 307-308 #1-15, 26-29