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5.10 Properties of Rhombuses, Rectangles, and Squares

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.

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5.10 Properties of Rhombuses, Rectangles, and Squares

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  1. 5.10 Properties of Rhombuses, Rectangles, and Squares

  2. 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.

  3. 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.

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

  5. Q Q a. S S 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. 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. SOLUTION

  6. Q b. R • If rhombus QRSTis a square, then all four angles are congruent right angles. So, if QRSTis a square. Because not all rhombuses are also squares, the statement is sometimes true. Q R 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. SOLUTION

  7. 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.

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