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Understanding Quadrilaterals: Rhombuses, Rectangles, and Squares

Explore the properties and corollaries of rhombuses, rectangles, and squares. Understand the conditions that define each type of quadrilateral and learn how to identify them. Practice using the theorems related to these special quadrilaterals through examples and explanations.

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Understanding Quadrilaterals: 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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