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6.3 Proving Quadrilaterals are Parallelograms Objective: Prove that a given quadrilateral is a parallelogram. Handbook, p. 19 6.3 Proving Quadrilaterals are Parallelograms If… then... both pairs of opposite sides are parallel (definition), >> one pair of opposite sides are BOTH parallel and congruent (6.3.1), >> > > it’s a parallelogram. 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 1 both consecutive angles (6.3.4),2 3 the diagonals bisect each other (6.3.5), m 2 m 1 180 m 2 m 3 180 it’s a parallelogram. it’s a parallelogram. 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. 6-3-1: One pair of opposite sides are parallel and congruent. 6-3-2: Both pairs of opposite sides are congruent 6-3-3: Both pairs of opposite angles are congruent. 6-3-4: One angle is supplementary to both consecutive angles. 6-3-5: The diagonals bisect each other. By 6-3-4, the quadrilateral must be a parallelogram. 6.3 Proving Quadrilaterals are Parallelograms Example 2: Determine if the quadrilateral must be a parallelogram. Justify your answer. Definition: Both pairs of opposite sides are parallel. 6-3-1: One pair of opposite sides are parallel and congruent. 6-3-2: Both pairs of opposite sides are congruent 6-3-3: Both pairs of opposite angles are congruent. 6-3-4: One angle is supplementary to both consecutive angles. 6-3-5: 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. Is there anything else we can add to our picture? Definition: Both pairs of opposite sides are parallel. 6-3-1: One pair of opposite sides are parallel and congruent. 6-3-2: Both pairs of opposite sides are congruent 6-3-3: Both pairs of opposite angles are congruent. 6-3-4: One angle is supplementary to both consecutive angles. 6-3-5: The diagonals bisect each other. 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. 6-3-1: One pair of opposite sides are parallel and congruent. 6-3-2: Both pairs of opposite sides are congruent 6-3-3: Both pairs of opposite angles are congruent. 6-3-4: One angle is supplementary to both consecutive angles. 6-3-5: 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. 6-3-1: One pair of opposite sides are parallel and congruent. 6-3-2: Both pairs of opposite sides are congruent 6-3-3: Both pairs of opposite angles are congruent. 6-3-4: One angle is supplementary to both consecutive angles. 6-3-5: The diagonals bisect each other. 15(3) 11 10(3) 4 34 34 5(9) 6 8(9) 21 51 51 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? 7(2.4) 2(2.4) 12 16.8 16.8 10(9) 16 9(9) 25 180 90 16 81 25 180 180 180 Definition: Both pairs of opposite sides are parallel. 6-3-1: One pair of opposite sides are parallel and congruent. 6-3-2: Both pairs of opposite sides are congruent 6-3-3: Both pairs of opposite angles are congruent. 6-3-4: One angle is supplementary to both consecutive angles. 6-3-5: 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 1 6 4 1 5 3 05 74 5 3 slope KL = slope JM 5 1 4 4 6 8 0 6 7 1 6 8 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 6-3-1 if A(2, 3), B(6, 2), C(5, 0), and D(1, 1). 6.3 Assignment p. 402: 9-23, 26