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Warm Up

Properties of Rhombuses, Rectangles, and Squares. Warm Up. Lesson Presentation. Lesson Quiz. =. =. =. 1. Give five ways to prove that a quadrilateral is a parallelogram. opp. sides , opp. sides , diags. bisects each other, opp. angles , a pair of opp. sides and. ANSWER.

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Warm Up

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  1. Properties of Rhombuses, Rectangles, and Squares Warm Up Lesson Presentation Lesson Quiz

  2. = = = 1. Give five ways to prove that a quadrilateral is a parallelogram. opp. sides , opp. sides , diags. bisects each other, opp. angles , a pair of opp. sides and ANSWER 2. Find xin the parallelogram. 14 ANSWER Warm-Up

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

  4. 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 For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. SOLUTION

  5. Example 2 Classify the special quadrilateral. Explain your reasoning. 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.

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

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

  8. 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.

  9. Example 3 Because ABCDis 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 ABCDare congruent.

  10. 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. ANSWER Q P R S Guided Practice 3. Sketch square PQRS. List everything you know about the square.

  11. Example 4 Carpentry You are building a frame for a window. The window will be installed in the opening shown in the diagram. 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 are54.8 inches and 55.3 inches. What can you conclude about the shape of the opening?

  12. 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. 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.

  13. ANSWER yes, Theorem 8.13 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.

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

  15. 3. Given rhombus MNOP, find m NPO. 48° ANSWER Lesson Quiz

  16. 4. Given rectangle QRST, find mRUS. 128° ANSWER Lesson Quiz

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