6-5 Conditions for Rhombuses, Rectangles, and Squares

Objective

To determine whether a parallelogram is a rhombus or rectangle

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Essential Understanding You can determine whether a parallelogram is a rhombus or a rectangle based on the properties of its diagonals.

Proof Proof of Theorem 6-16

Given: ABCD is a parallelogram, A C ¯ ⊥ B D ¯ eh c bar , up tack , b d bar

Prove: ABCD is a rhombus.

Since ABCD is a parallelogram, A C ¯ eh c bar and B D ¯ b d bar bisect each other, so B E ¯ ≅ D E ¯ . b e bar , approximately equal to , d e bar , . Since A C ¯ ⊥ B D ¯ , ∠ A E D eh c bar , up tack , b d bar , comma . angle eh e d and ∠ A E B angle eh e b are congruent right angles. By the Reflexive Property of Congruence, A E ¯ ≅ A E ¯ . eh e bar , approximately equal to , eh e bar , . So Δ A E B ≅ Δ A E D cap delta eh e b approximately equal to cap delta eh e d by SAS. Corresponding parts of congruent triangles are congruent, so A B ¯ ≅ A D ¯ . eh b bar , approximately equal to , eh d bar , . Since opposite sides of a parallelogram are congruent, A B ¯ ≅ D C ¯ ≅ B C ¯ ≅ A D ¯ . eh b bar , approximately equal to , d c bar , approximately equal to , b c bar , approximately equal to , eh d bar , . By definition, ABCD is a rhombus.

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