Objective: Prove that a given quadrilateral is a parallelogram. 6.3 Proving Quadrilaterals are Parallelograms Handbook, p. 19

6.3 Proving Quadrilaterals are Parallelograms If…then... both pairs of opposite sides are parallel (definition), it’s a parallelogram. one pair of opposite sides are BOTH parallel and congruent (6.3.1), it’s a parallelogram. both pairs of opposite sides are congruent (6.3.2), it’s a parallelogram. both pairs of opposite angles are congruent (6.3.3), it’s a parallelogram. one angle is supplementary to both consecutive angles (6.3.4), it’s a parallelogram. the diagonals bisect each other (6.3.5), it’s a parallelogram. >> > > > > 1 3 2

6.3 Proving Quadrilaterals are Parallelograms Example 1: Determine if the quadrilateral must be a parallelogram. Justify your answer. Definition: Both pairs of opposite sides are parallel : One pair of opposite sides are parallel and congruent : Both pairs of opposite sides are congruent 6-3-3: Both pairs of opposite angles are congruent : One angle is supplementary to both consecutive angles : The diagonals bisect each other. By 6-3-4, the quadrilateral must be a parallelogram.

Example 2: Determine if the quadrilateral must be a parallelogram. Justify your answer. 6.3 Proving Quadrilaterals are Parallelograms Definition: Both pairs of opposite sides are parallel : One pair of opposite sides are parallel and congruent : Both pairs of opposite sides are congruent 6-3-3: Both pairs of opposite angles are congruent : One angle is supplementary to both consecutive angles : The diagonals bisect each other. Only one pair of opposite angles are congruent, so not enough information.

6.3 Proving Quadrilaterals are Parallelograms Example 3: Determine if the quadrilateral must be a parallelogram. Justify your answer. Definition: Both pairs of opposite sides are parallel : One pair of opposite sides are parallel and congruent : Both pairs of opposite sides are congruent 6-3-3: Both pairs of opposite angles are congruent : One angle is supplementary to both consecutive angles : The diagonals bisect each other. Is there anything else we can add to our picture? Both pairs of opposite angles are congruent, so the quadrilateral must be a parallelogram.

6.3 Proving Quadrilaterals are Parallelograms Example 4: Determine if the quadrilateral must be a parallelogram. Justify your answer. Definition: Both pairs of opposite sides are parallel : One pair of opposite sides are parallel and congruent : Both pairs of opposite sides are congruent 6-3-3: Both pairs of opposite angles are congruent : One angle is supplementary to both consecutive angles : The diagonals bisect each other. Consecutive sides, not opposite sides are marked congruent, so the quadrilateral IS NOT a parallelogram.

6.3 Proving Quadrilaterals are Parallelograms Example 5: Show that JKLM is a parallelogram for a = 3 and b = 9. Definition: Both pairs of opposite sides are parallel : One pair of opposite sides are parallel and congruent : Both pairs of opposite sides are congruent 6-3-3: Both pairs of opposite angles are congruent : One angle is supplementary to both consecutive angles : The diagonals bisect each other. Since both pairs of opposite sides are congruent JKLM is a parallelogram.

6.3 Proving Quadrilaterals are Parallelograms Example 6: Show that PQRS is a parallelogram for a = 2.4 and b = 9. What will a and b let me find? Definition: Both pairs of opposite sides are parallel : One pair of opposite sides are parallel and congruent : Both pairs of opposite sides are congruent 6-3-3: Both pairs of opposite angles are congruent : One angle is supplementary to both consecutive angles : The diagonals bisect each other. Since one pair of opposite sides are both parallel and congruent, PQRS is a parallelogram.

6.3 Proving Quadrilaterals are Parallelograms Example 7: Show that quadrilateral JKLM is a parallelogram by using the definition of parallelogram. J(–1, –6), K(–4, –1), L(4, 5), M(7, 0). Definition: Opposite sides are parallel, so we need to show: slope JK = slope LM slope KL = slope JM Since slopes of opposite sides are the same, the opposite sides are parallel.

6.3 Proving Quadrilaterals are Parallelograms Example 8: Show that quadrilateral ABCD is a parallelogram by using Theorem if A(2, 3), B(6, 2), C(5, 0), and D(1, 1).

6.3 Assignment p. 402: 9-23, 26