Ways to Prove that Quadrilaterals are Parallelograms Keystone Geometry

Remember! »Definition of a parallelogram: A quadrilateral with both pairs of opposite sides parallel.

3 Properties »Opposite sides are congruent. »Opposite angles are congruent. »Diagonals bisect each other, but are not necessarily congruent.

»If you have a quadrilateral and want to prove that it is a parallelogram you need to show that: Both pairs of opposite sides are parallel.

4 Theorems 1. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram

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

3. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

4. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Summary »If we KNOW the quadrilateral is a parallelogram: Definition: Opp. Sides are parallel Prop. 1: Opp. Sides are congruent Prop. 2: Opposite Angles are Congruent Prop. 3: Diagonals bisect each other »If we want to PROVE the quadrilateral is a parallelogram: Thm. 1: Both pairs of opposite are congruent (Prop. 1) Them. 2: One pair of opposite sides are BOTH congruent and parallel (Combo of Def./Prop. 1) Thm. 3: Both pairs of opposite angles are congruent (Prop. 2) Them 4: Diagonals bisect each other (Prop. 3) »If we want to PROVE the quadrilateral is a parallelogram: Thm. 1: Both pairs of opposite are congruent (Prop. 1) Them. 2: One pair of opposite sides are BOTH congruent and parallel (Combo of Def./Prop. 1) Thm. 3: Both pairs of opposite angles are congruent (Prop. 2) Them 4: Diagonals bisect each other (Prop. 3)