Lesson 8 3 Tests for Parallelograms Proving Quadrilaterals

Proving Quadrilaterals as Parallelograms Theorem 1: If both pairs of opposite sides of a

Theorem: Theorem 3: If both pairs of opposite angles of a quadrilateral are congruent,

5 ways to prove that a quadrilateral is a parallelogram. 1. Show that both

Examples …… Example 1: Find the value of x and y that ensures the

Examples …… Example 3: Find the value of x and y that ensures the

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Lesson 8 -3 Tests for Parallelograms

Proving Quadrilaterals as Parallelograms Theorem 1: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. H Theorem 2: G E F If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.

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

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.

Examples …… Example 1: Find the value of x and y that ensures the quadrilateral y+2 is a parallelogram. 6 x 4 x+8 2 y Example 2: Find the value of x and y that ensure the quadrilateral is a parallelogram. (2 x + 8)° 120° 5 y°

Examples …… Example 3: Find the value of x and y that ensures the quadrilateral 5 x + 12 is a parallelogram. 3 y + 6 5 y-6 8 x Example 4: Find the value of x and y that ensure the quadrilateral is a parallelogram. 4 x 2 y y+7 5 x - 4