 5.10 Properties of Rhombuses, Rectangles, and Squares

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5.10 Properties of Rhombuses, Rectangles, and Squares

Vocabulary A rhombus is a parallelogram with four congruent sides.
A rectangle is a parallelogram with four right angles. A square is a parallelogram with four congruent sides and four right angles.

Corollaries Rhombus Corollary: A quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle Corollary: A quadrilateral is a rectangle if and only if it has four right angles. Square Corollary: A quadrilateral is a square if and only if it is a rhombus and a rectangle.

ABCD is a rhombus. ABCD is a rectangle. ABCD is a square.

EXAMPLE 1 Use properties of special quadrilaterals For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. a. Q S SOLUTION a. By definition, a rhombus is a parallelogram with four congruent sides. By Theorem 8.4, opposite angles of a parallelogram are congruent. So, The statement is always true. Q S

EXAMPLE 1 Use properties of special quadrilaterals For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. Q R b. SOLUTION If rhombus QRST is a square, then all four angles are congruent right angles. So, if QRST is a square. Because not all rhombuses are also squares, the statement is sometimes true. Q R

Theorem 5.26: A parallelogram is a rhombus if and only if its diagonals are perpendicular. Theorem 5.27: A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. Theorem 5.28: A parallelogram is a rectangle if and only if its diagonals are congruent.