8.4 properties of rhombuses, rectangles, and squares

1. 8.4 Properties of Rhombuses , Rectangles, and Squares Bell Thinger ANSWER 14 1. Give five ways to prove that a quadrilateral is a parallelogram. ANSWER opp. sides , opp. sides , diags. bisects each other, opp. angles , a pair of opp. sides and = = = 2. Find x in the parallelogram.

2. Example 1 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

3. Example 1 For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. 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 Q R b. SOLUTION

4. Example 2 SOLUTION Classify the special quadrilateral. Explain your reasoning. 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.

5. Guided Practice 1. For any rectangle EFGH , is it always or sometimes true that Explain your reasoning. FG GH ? Sometimes; this is only true if EFGH is a square. ANSWER

6. Guided Practice 2. A quadrilateral has four congruent sides and four congruent angles. Sketch the quadrilateral and classify it. ANSWER square

7. Example 3 Sketch rectangle ABCD . List everything that you know about it. SOLUTION By definition, you need to draw a figure with the following properties: • The figure is a parallelogram. • The figure has four right angles.

8. Example 3 Because ABCD is a parallelogram, it also has these properties: • Opposite sides are parallel and congruent. • Opposite angles are congruent. Consecutive angles are supplementary. • Diagonals bisect each other. By Theorem 8.13 , the diagonals of ABCD are congruent.

9. Guided Practice 3. Sketch square PQRS . List everything you know about the square. ANSWER P Q S R PQRS is a parallelogram, rectangle and a rhombus. Opposite pairs of sides are parallel and all four sides are congruent. All four angles are right angles. Diagonals are congruent and bisect each other. Diagonals are perpendicular and each diagonal bisects a pair of opposite angles.

10. Example 4 Carpentry a. The opening must be a rectangle. Given the measurements in the diagram, can you assume that it is? Explain . b. You measure the diagonals of the opening. The diagonals are 54.8 inches and 55.3 inches. What can you conclude about the shape of the opening? You are building a frame for a window. The window will be installed in the opening shown in the diagram.

11. Example 4 SOLUTION b. By Theorem 8.13 , the diagonals of a rectangle are congruent. The diagonals of the quadrilateral formed by the boards are not congruent, so the boards do not form a rectangle. No, you cannot. The boards on opposite sides are the same length, so they form a parallelogram. But you do not know whether the angles are right angles. a.

12. Guided Practice 4. Suppose you measure only the diagonals of a window opening. If the diagonals have the same measure, can you conclude that the opening is a rectangle? Explain . ANSWER yes, Theorem 8.13

13. Exit Slip Name each type of quadrilateral (parallelogram, rectangle, rhombus, and square) for which the statement is true. 1. Both pairs of angles are congruent. ANSWER parallelogram, rectangle, rhombus, square 2. The quadrilateral is equilateral. ANSWER rhombus, square

14. Exit Slip 3. Given rhombus MNOP, find m NPO . ANSWER 48 °

15. Exit Slip 4. Given rectangle QRST, find m RUS . ANSWER 128 °

16. Pg 557-560 #17, 24, 26, 29, 34