1. Proving Quadrilaterals are Parallelograms

2. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 1 H G E F

3. Theorem 2 If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram. H FE G

4. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 3 H G E F then Quad. EFGH is a parallelogram.

5. Theorem 4 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. then Quad. EFGH is a parallelogram. EM = GM and HM = FM M FE H G

6. 5 ways to prove that a quadrilateral is a parallelogram. 1. Show that both pairs of opposite sides are ||. [definition] 2. Show that both pairs of opposite sides are . 3. Show that one pair of opposite sides are both  and ||. 4. Show that both pairs of opposite angles are . 5. Show that the diagonals bisect each other.

7. Examples …… Find the value of x and y that ensures the quadrilateral is a parallelogram. Example 1: 6x 4x+8 y+2 2y 6x = 4x+8 2x = 8 x = 4 units 2y = y+2 y = 2 units Example 2: Find the value of x and y that ensure the quadrilateral is a parallelogram. 120° 5y° (2x + 8)° 2x + 8 = 120 2x = 112 x = 56 units 5y + 120 = 180 5y = 60 y = 12 units