1. 6-3 Proving a Quad is a parallelogram

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

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

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

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

6. 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.

7. 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)

8. 1. No, 2. Yes, 3. No

9. Practice
Pg #1-15, 26-29