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6.3 Proving Quadrilaterals are Parallelograms

Learn how to prove if a given quadrilateral is a parallelogram through examples and justifications based on the properties of parallel sides and angles. Explore different scenarios and apply the criteria to determine parallelograms.

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6.3 Proving Quadrilaterals are Parallelograms

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  1. 6.3 Proving Quadrilaterals are Parallelograms Objective: Prove that a given quadrilateral is a parallelogram. Handbook, p. 19

  2. >> 6.3 Proving Quadrilaterals are Parallelograms > > >> > > 1 2 3

  3. 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.

  4. 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.

  5. 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. 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.

  7. 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. Since both pairs of opposite sides are congruent JKLM is a parallelogram.

  8. 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. • 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.

  9. 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.

  10. 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).

  11. 6.3 Assignment p. 402: 9-23, 26

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