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Understanding Rhombuses and Their Properties in Quadrilaterals

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A rhombus is defined as a parallelogram with four congruent sides. According to Theorem 8.4, opposite angles of a parallelogram are congruent, thus the property holds true for rhombuses. Through examples, we explore whether certain statements about rhombuses are always or sometimes true. For instance, while a rhombus may appear as a square with all angles being right angles, not all rhombuses are squares. This guide will help classify quadrilaterals based on their properties and clarify reasoning through sketches and explanations.

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Understanding Rhombuses and Their Properties in Quadrilaterals

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  1. 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

  2. Q Q b. R 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 b. 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.

  3. Classify the special quadrilateral. Explain your reasoning. EXAMPLE 2 Classify special quadrilaterals SOLUTION The quadrilateral has four congruent sides. One of the angles is not a right angle, so the rhombus is not also a square. By the Rhombus Corollary, the quadrilateral is a rhombus.

  4. 1. For any rectangle EFGH, is it always or sometimes true that Explain your reasoning. FG FG GH ? GH ANSWER Adjacent sides of a rectangle can be congruent . If, it is a square. A square is also a rectangle with four right angles but rectangle is not always a square. Therefore , in EFGH , only if EFGH is a square. for Examples 1 and 2 GUIDED PRACTICE

  5. ANSWER D C Square A B for Examples 1 and 2 GUIDED PRACTICE 2. A quadrilateral has four congruent sides and four congruent angles. Sketch the quadrilateral and classify it.

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