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NAME ______________________________________________ DATE 6-5 ____________ PERIOD _____ Lesson Reading Guide Rhombi and Squares Get Ready for the Lesson Read the introduction to Lesson 6-5 in your textbook. If you draw a diagonal on the surface of one of the square wheels shown in the picture in your textbook, how can you describe the two triangles that are formed? Read the Lesson 1. Sketch each of the following. a. a quadrilateral with perpendicular diagonals that is not a rhombus b. a quadrilateral with congruent diagonals that is not a rectangle c. a quadrilateral whose diagonals are perpendicular and bisect each other, but are not congruent a. The diagonals are congruent. b. Opposite sides are congruent. c. The diagonals are perpendicular. d. Consecutive angles are supplementary. e. The quadrilateral is equilateral. f. The quadrilateral is equiangular. g. The diagonals are perpendicular and congruent. h. A pair of opposite sides is both parallel and congruent. 3. What is the common name for a regular quadrilateral? Explain your answer. Remember What You Learned 4. A good way to remember something is to explain it to someone else. Suppose that your classmate Luis is having trouble remembering which of the properties he has learned in this chapter apply to squares. How can you help him? Chapter 6 35 Glencoe Geometry Lesson 6-5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 2. List all of the following special quadrilaterals that have each listed property: parallelogram, rectangle, rhombus, square. NAME ______________________________________________ DATE 6-5 ____________ PERIOD _____ Study Guide and Intervention Rhombi and Squares Properties of Rhombi A rhombus is a quadrilateral with four congruent sides. Opposite sides are congruent, so a rhombus is also a parallelogram and has all of the properties of a parallelogram. Rhombi also have the following properties. R H B M The diagonals are perpendicular. 苶 MH 苶⊥苶 RO 苶 Each diagonal bisects a pair of opposite angles. 苶 MH 苶 bisects ⬔RMO and ⬔RHO. 苶O R 苶 bisects ⬔MRH and ⬔MOH. If the diagonals of a parallelogram are perpendicular, then the figure is a rhombus. 苶O 苶⊥苶 MH 苶, If RHOM is a parallelogram and R then RHOM is a rhombus. O Example In rhombus ABCD, m⬔BAC ⫽ 32. Find the measure of each numbered angle. ABCD is a rhombus, so the diagonals are perpendicular and 䉭ABE is a right triangle. Thus m⬔4 ⫽ 90 and m⬔1 ⫽ 90 ⫺ 32 or 58. The diagonals in a rhombus bisect the vertex angles, so m⬔1 ⫽ m⬔2. Thus, m⬔2 ⫽ 58. B 12 32⬚ 4 A E 3 C D A rhombus is a parallelogram, so the opposite sides are parallel. ⬔BAC and ⬔3 are alternate interior angles for parallel lines, so m⬔3 ⫽ 32. Exercises ABCD is a rhombus. B C 1. If m⬔ABD ⫽ 60, find m⬔BDC. 2. If AE ⫽ 8, find AC. 3. If AB ⫽ 26 and BD ⫽ 20, find AE. 4. Find m⬔CEB. 5. If m⬔CBD ⫽ 58, find m⬔ACB. 6. If AE ⫽ 3x ⫺ 1 and AC ⫽ 16, find x. A D 7. If m⬔CDB ⫽ 6y and m⬔ACB ⫽ 2y ⫹ 10, find y. 8. If AD ⫽ 2x ⫹ 4 and CD ⫽ 4x ⫺ 4, find x. 9. a. What is the midpoint of 苶 FH 苶? y J H 苶? b. What is the midpoint of 苶 GJ c. What kind of figure is FGHJ? Explain. F O 苶H 苶? d. What is the slope of F G x e. What is the slope of G 苶J 苶? f. Based on parts c, d, and e, what kind of figure is FGHJ ? Explain. Chapter 6 36 Glencoe Geometry Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. E NAME ______________________________________________ DATE 6-5 Study Guide and Intervention ____________ PERIOD _____ (continued) Rhombi and Squares Properties of Squares A square has all the properties of a rhombus and all the properties of a rectangle. Example Find the measure of each numbered angle of B square ABCD. C 1 Using properties of rhombi and rectangles, the diagonals are perpendicular and congruent. 䉭ABE is a right triangle, so m⬔1 ⫽ m⬔2 ⫽ 90. A E 5 2 4 3 D Each vertex angle is a right angle and the diagonals bisect the vertex angles, so m⬔3 ⫽ m⬔4 ⫽ m⬔5 ⫽ 45. Exercises Determine whether the given vertices represent a parallelogram, rectangle, rhombus, or square. Explain your reasoning. 1. A(0, 2), B(2, 4), C(4, 2), D(2, 0) 3. A(⫺2, ⫺1), B(0, 2), C(2, ⫺1), D(0, ⫺4) 4. A(⫺3, 0), B(⫺1, 3), C(5, ⫺1), D(3, ⫺4) 5. S(⫺1, 4), T(3, 2), U(1, ⫺2), V(⫺3, 0) 6. F(⫺1, 0), G(1, 3), H(4, 1), I(2, ⫺2) 7. Square RSTU has vertices R(⫺3, ⫺1), S(⫺1, 2), and T(2, 0). Find the coordinates of vertex U. Chapter 6 37 Glencoe Geometry Lesson 6-5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 2. D(⫺2, 1), E(⫺1, 3), F(3, 1), G(2, ⫺1)