Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Rational trigonometry wikipedia , lookup
Noether's theorem wikipedia , lookup
Riemann–Roch theorem wikipedia , lookup
Trigonometric functions wikipedia , lookup
History of geometry wikipedia , lookup
Euler angles wikipedia , lookup
History of trigonometry wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Four color theorem wikipedia , lookup
6-4 6-4 Special Parallelograms 1. Plan What You’ll Learn Check Skills You’ll Need • To use properties of PACE is a parallelogram and mlPAC ≠ 109. Complete each of the following. diagonals of rhombuses and rectangles • To determine whether a parallelogram is a rhombus or a rectangle . . . And Why Lesson 6-2 4.75 4.5 R 6. CP = j 9.5 4 Finding Angle Measures Finding Diagonal Lengths Identifying Special Parallelograms Real-World Connection The definitions in Lesson 6-1 of rhombus, rectangle, and square included more information than necessary. Definitions with less restrictive conditions can often be used. For example, a rectangle can be defined as a parallelogram with one right angle. The fact that it has four right angles follows from the properties of a parallelogram. Using a definition with less restrictive conditions makes establishing sufficient conditions easier. R P Q Theorem 6-9 Each diagonal of a rhombus bisects two angles of the rhombus. Proof of Theorem 6-9 More Math Background: p. 304D Given: rhombus ABCD B Prove: AC bisects &BAD and &BCD. A segment bisects an angle if and only if it divides the angle into two congruent angles. 1 2 3 Math Background S Vocabulary Tip To use properties of diagonals of rhombuses and rectangles To determine whether a parallelogram is a rhombus or a rectangle Examples E 9. Draw a rhombus that is not a square. Draw a rectangle that is not a square. Explain why each is not a square. See back of book. If you draw two congruent isosceles triangles with base PQ, you have drawn a rhombus. Note that &RPQ, &SPQ, &RQP, and &SQP are all congruent. This suggests Theorem 6-9 and its proof. Proof 1 2 C Diagonals of Rhombuses and Rectangles Key Concepts Objectives 3.5 3. m&CEP = j 109 4. PR = j 4.75 5. RE = j 3.5 7 A 2. EP = j 7 7. m&EPA = j 71 8. m&ECA = j 71 P To lay out a rectangular patio, as in Example 4 1 1. EC = j 4.5 GO for Help 3 4 C Lesson Planning and Resources 1 Proof: ABCD is a rhombus, so its sides are 2 A D all congruent. AC > AC by the Reflexive Property of Congruence. Therefore, #ABC > #ADC by the SSS Postulate. &1 > &2 and &3 > &4 by CPCTC. Therefore, AC bisects &BAD and &BCD by the definition of bisect. See p. 304E for a list of the resources that support this lesson. You can show similarly that BD bisects &ABC and &ADC. PowerPoint Bell Ringer Practice The diagonals of a rhombus provide an interesting application of the Converse of the Perpendicular Bisector Theorem. Check Skills You’ll Need For intervention, direct students to: Properties: Sides and Angles Lesson 6-4 Special Parallelograms Special Needs Below Level L1 Some students think that because the sides of a rhombus are congruent its diagonals must also be congruent. Draw or have students draw several counterexamples. learning style: visual 329 L2 Have students use paper folding to explore and explain Theorems 6-9 and 6-10. learning style: tactile Lesson 6-2: Examples 1, 2 Extra Skills, Word Problems, Proof Practice, Ch. 6 Properties: Diagonals and Transversals Lesson 6-2: Example 3 Extra Skills, Word Problems, Proof Practice, Ch. 6 329 2. Teach In the rhombus at the right, points B and D are equidistant from A and C. By the Converse of the Perpendicular Bisector Theorem, they are on the perpendicular bisector of AC. This proves the next theorem. In Exercise 22, you will prove it a second way. Guided Instruction Key Concepts Alternate Method Challenge students to suggest a way to prove that each diagonal of a rhombus bisects two angles of the rhombus without using congruent triangles. They could use the Isosceles Triangle Theorem and a property of parallel lines to show that &1 &2 &3 &4. 1 EXAMPLE 3 A 2 A D Finding Angle Measures EXAMPLE MNPQ is a rhombus and m&N = 120. Find the measures of the numbered angles. N m&1 = m&3 Isosceles Triangle Theorem Real-World m&1 + m&3 + 120 = 180 Connection The diagonals of the rhombus formed by the pantograph stay perpendicular when the pantograph lifts or lowers. Quick Check 2(m&1) + 120 = 180 2(m&1) = 60 m&1 = 30 1 Triangle Angle-Sum Theorem P 2 Q M Substitute. 3 4 120 Subtract 120 from each side. Divide each side by 2. Therefore, m&1 = m&3 = 30. By Theorem 6-9, m&1 = m&2 and m&3 = m&4. Therefore, m&1 = m&2 = m&3 = m&4 = 30. 50 1 1 Find the measures of the numbered angles in the rhombus. l1 ≠ 90, l2 ≠ 50, l3 ≠ 50, l4 ≠ 40 C 1 C You can use Theorems 6-9 and 6-10 to find angle measures in rhombuses. 1 Find the measures of the numbered angles in the rhombus. 78° D B AC ' BD pantograph Additional Examples B C The diagonals of a rhombus are perpendicular. 1 PowerPoint A Theorem 6-10 Teaching Tip In Quick Check, point out that the diagonals form eight angles at the vertices of the rhombus. When the measure of one angle is known, the other seven angle measures can be found using properties of a rhombus. B 4 2 3 The diagonals of a rectangle, another parallelogram, also have a special property. 4 D ml1 ≠ 78, ml2 ≠ 90, ml3 ≠ 12, ml4 ≠ 78 Key Concepts Theorem 6-11 The diagonals of a rectangle are congruent. 2 One diagonal of a rectangle has length 8x + 2. The other diagonal has length 5x + 11. Find the length of each diagonal. 26, 26 Proof Proof of Theorem 6-11 Given: Rectangle ABCD Prove: AC > BD A D B C Proof: ABCD is a rectangle, so it is also a parallelogram. AB > DC because opposite sides of a parallelogram are congruent. BC > BC by the Reflexive Property of Congruence. &ABC and &DCB are right angles by the definition of rectangle. &ABC > &DCB because all right angles are congruent. #ABC > #DCB by SAS. AC > BD by CPCTC. 330 Chapter 6 Quadrilaterals Advanced Learners English Language Learners ELL L4 Have students rewrite Theorems 6-12, 6-13, and 6-14 so that the parallelogram must be a square. 330 learning style: verbal Review the terminology of conditionals by examining Theorems 6-12, 6-13, and 6-14. Point out that each hypothesis contains two conditions, the first in each being that the figure is a parallelogram. learning style: verbal Guided Instruction 4 5 A A D B B E D C C Finding Diagonal Length E D C B EXAMPLE E D C B A 3 C B A 2 C B A 1 2 E D E D E Test-Taking Tip Because FD 5 GE, you can substitute the value of y into the expression for GE to check your work. Multiple Choice Find the length of FD in rectangle GFED if FD = 2y + 4 and GE = 6y - 5. 2 12 3 14 4 12 8 12 F E 2y + 4 = 6y - 5 G D Diagonals of a rectangle are congruent. 9 = 4y Subtract 2y from each side and add 5 to each side. 9 =y 4 Divide each side by 4. Theorems 6-12, 6-13, and 6-14 can be proved as class exercises when they are presented. Encourage students to suggest as many plans as they can to prove each theorem. 3 1 FD = 2 Q 94 R + 4 = 17 2 , or 8 2. The correct answer is D. Quick Check Teaching Tip 2 Find the length of the diagonals of GFED if FD = 5y - 9 and GE = y + 5. 8 1 2 Visual Learners Students may wish to try drawing the figures described before answering the questions. 4 2 1 EXAMPLE EXAMPLE Careers Making sure that an angle is right or that a line is straight is necessary for a carpenter building the frame of a house or a surveyor marking off a plot of land. Is the Parallelogram a Rhombus or a Rectangle? The following theorems are the converses of Theorems 6-9, 6-10, and 6-11, respectively. You will prove these theorems in the Exercises. PowerPoint Key Concepts Theorem 6-12 Additional Examples If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. Theorem 6-13 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. Theorem 6-14 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. 4 Explain how you could use the properties of diagonals to stake the vertices of a play area shaped like a rhombus. Sample: Position ropes at right angles to each other at their midpoints, and stake the endpoints. You can use Theorems 6-12, 6-13, and 6-14 to classify parallelograms. 3 EXAMPLE 3 The diagonals of ABCD are perpendicular. AB = 16 cm and BC = 8 cm. Can ABCD be a rhombus or rectangle? Explain. No; perpendicular diagonals in a parallelogram mean that the figure is a rhombus, but ABCD is not a rhombus because its side lengths are not equal. Identifying Special Parallelograms Can you conclude that the parallelogram is a rhombus or a rectangle? Explain. a. b. A D For: Quadrilateral Activity Use: Interactive Textbook, 6-4 Quick Check Yes. A diagonal bisects two angles. By Theorem 6-12, this parallelogram is a rhombus. Resources • Daily Notetaking Guide 6-4 L3 • Daily Notetaking Guide 6-4— L1 Adapted Instruction B AC BD C Yes. The diagonals are congruent. By Theorem 6-14, this parallelogram is a rectangle. 3 A parallelogram has angles of 30°, 150°, 30°, and 150°. Can you conclude that it is a rhombus or a rectangle? Explain. Sample: No; there is not enough information to conclude that the parallelogram is a rhombus. It cannot be a rectangle because it has no right angles. Lesson 6-4 Special Parallelograms 331 Closure A quadrilateral has congruent diagonals. Explain why it may or may not be a rectangle. Sample: If congruent diagonals bisect each other, the figure must be a rectangle by Theorem 6-14. If they do not bisect each other, the quadrilateral is not even a parallelogram, so it is not a rectangle by the contrapositive of Theorem 6-3. 331 3. Practice You can use properties of diagonals to construct special parallelograms. 4 Assignment Guide Connection Community Service Builders use properties of diagonals to “square up” rectangular shapes like building frames and playing-field boundaries. 1 A B 1-15, 22, 44-46, 50-59 2 A B 16-21, 23-43, 47-49 C Challenge 60-63 Test Prep Mixed Review Real-World EXAMPLE Suppose you are on the volunteer building team at the right. You are helping to lay out a rectangular patio. Explain how to use properties of diagonals to locate the four corners. 64-66 67-89 To locate the corners, you can use two theorems: Homework Quick Check • Theorem 6-7: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. To check students’ understanding of key skills and concepts, go over Exercises 6, 20, 23, 38, 46. Exercises 1–9 Have students state which property of a rhombus they use to find each angle measure. • Theorem 6-14: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Real-World Connection A well-planned volunteer effort can frame a small house in a day. Exercises 10–15 Each exercise Quick Check requires students to remember that the diagonals of a rectangle are congruent. Before having students begin this exercise set, make sure that they recall this property. First, cut two pieces of rope that will be the diagonals of the foundation rectangle. Cut them the same length because of Theorem 6-14. Join them at their midpoints because of Theorem 6-7. Then pull the ropes straight and taut. The ends of the ropes will be the corners of a rectangle. 4 Kate thinks that they can adapt this method slightly to stake off a square play area. Is she right? Explain. Yes; if the ropes are # to each other, then the endpoints of the ropes determine a square. EXERCISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. Practice and Problem Solving A Practice by Example Find the measures of the numbered angles in each rhombus. 1. Example 1 GO for Help GPS Guided Problem Solving L3 L4 Enrichment L2 Reteaching Practice Name Class 1 4. 4. 33.5, 33.5, 113, 33.5 5. 32, 90, 58, 32 Special Parallelograms 2. 7. 54ⴗ 1 4 4 2 3 4. 1 4 30ⴗ 4 2 3 2 4 9 60ⴗ 1 2 3 4 68ⴗ 8. 9. 5 (page 331) 3.5 9 9 3.5 9 5 10–15. See margin. 10. HJ = x and IK = 2x - 7 11. HJ = 3x + 5 and IK = 5x - 9 12. HJ = 3x + 7 and IK = 6x - 11 13. HJ = 19 + 2x and IK = 3x + 22 For each rhombus, (a) find the measures of the numbered angles, and then (b) find the area. 15. AC = 8 in. BD = 22 in. 16 cm 29ⴗ 9 cm 1 2 3 20ⴗ 4 D 16. A 1 2 52ⴗ B 1 3 4 C 10 m 2 3 13 m 4 © Pearson Education, Inc. All rights reserved. HIJK is a rectangle. Find the value of x and the length of each diagonal. 14. 2 18. Opposite angles are congruent and supplementary, but the quadrilateral is not a rectangle. 332 4 3 2 1 58⬚ 8. 30⬚ 9. 2 3 2 1 2 3 35⬚ 90, 55, 90 332 10. LN = x and MP = 2x - 4 11. LN = 5x - 8 and MP = 2x + 1 12. LN = 3x + 1 and MP = 8x - 4 13. LN = 9x - 14 and MP = 7x + 4 14. LN = 7x - 2 and MP = 4x + 3 15. LN = 3x + 5 and MP = 9x - 10 Chapter 6 Quadrilaterals Determine whether the quadrilateral can be a parallelogram. If not, write impossible. Explain your answer. 17. One pair of opposite sides is parallel, and the other pair is congruent. 6. 3 1 60⬚ 35⬚ 1 60, 90, 30 55, 35, 55, 90 2 Example 2 x Algebra LMNP is a rectangle. Find the value of x and the length of each diagonal. 2 6. 3 The parallelograms below are not drawn to scale. Can the parallelogram have the conditions marked? If not, write impossible. Explain your answer. 7. 5. 4 3 118, 31, 31 26, 128, 128 106ⴗ 3 3 1 5. 1 1 59ⴗ 2 3 1 118⬚ 3 113⬚ 4 3. 18ⴗ 3. 26⬚ 1 4 1 For each parallelogram, (a) choose the best name, and then (b) find the measures of the numbered angles. 1. 2 3 1 2 2 104⬚ 2 L3 Date Practice 6-4 2. 4 38, 38, 38, 38 6. 90, 60, 60, 30 L1 Adapted Practice 3 (page 330) 10. 4; LN ≠ MP ≠ 4 13. 9; LN ≠ MP ≠ 67 11. 3; LN ≠ MP ≠ 7 2 14. 35 ; LN ≠ MP ≠ 29 3 ≠ 93 12. 1; LN ≠ MP ≠ 4 15. 52 ; LN ≠ MP ≠ 12 12 Diversity Is the parallelogram a rhombus or a rectangle? Justify your answer. Example 3 (page 331) 16. 17. Exercise 19 Depending on where students live, older screen doors may not be familiar sights. 18. Exercise 21 Ask: What is a name for a rectangle that is also a rhombus? square 16–18. See back of book. 19. Hardware You can use a simple device called a turnbuckle to “square up” structures that are parallelograms. For the gate pictured at the right, you tighten or loosen the turnbuckle on the diagonal cable so that the cable stays congruent to the other diagonal. Explain why a frame that normally is rectangular will, when it sags, keep the shape of a parallelogram. See margin. 20. Carpentry A carpenter is building a bookcase. How can she use a tape measure to check that the bookshelf is rectangular? Justify your answer and name any theorems used. See margin. 21. Reasoning Suppose the diagonals of a parallelogram are both perpendicular and Turnbuckle congruent. What type of special quadrilateral is it? Explain your reasoning. See back of book. Example 4 (page 332) B Apply Your Skills Exercise 24 Some students may select answer choice C for square because it satisfies the condition. Call attention to the term must which indicates that the most general answer possible, choice A rhombus, is best. Point out that a square is a rhombus but a rhombus is not necessarily a square. Exercises 35, 36 Suggest that students add symbols for trapezoids and kites after the appropriate properties in Exercises 25–34 to summarize all their work in Chapter 6. 22. Developing Proof Complete the flow proof of Theorem 6-10. 22a. Def. of a rhombus b. Diagonals of a ~ bisect each other. c. AE O AE d. Reflexive Prop. of O e. kABE O kADE f. CPCTC g. l Add. Post. h. lAEB and lAED are rt. '. i. O suppl. ' are rt. ' Thm. j. Def. of # Given: ABCD is a rhombus. C E A the methods for solving a system of equations. ABCD is a rhombus. Given Prove: AC ' BD B Exercise 46 If necessary, review AB AD a. 9 D BE DE b. 9 c. 9 d. 9 e. 9 SSS m⬔AEB + m⬔AED =180 g. 9 ⬔AEB ⬔AED f. 9 19. The pairs of opp. sides of the frame remain O, so the frame remains a ~. h. 9 i. 9 20. After measuring the sides, she can measure the diagonals. AC # BD j. 9 Proof 23. Prove Theorem 6-13. Given: $ABCD; AC ' BD at E. GO nline Homework Help Visit: PHSchool.com Web Code: aue-0604 B C E Prove: ABCD is a rhombus. A D See margin. 24. Multiple Choice A diagonal of a parallelogram bisects one angle of the parallelogram. What kind of quadrilateral must the figure be? A rhombus rectangle square cannot tell Lesson 6-4 Special Parallelograms 333 23. Answers may vary. Sample: The diagonals of a ~ bisect each other so AE O CE. Both lAED and lCED are right ' because AC # BD, and since DE O DE by the Reflexive Prop., kAED O kCED by SAS. By CPCTC AD O CD, and since opp. sides of a ~ are O, AB O BC O CD O AD. So ABCD is a rhombus because it has 4 O sides. 333 4. Assess & Reteach Using Symbols Create your own distinctive symbols for a parallelogram, rhombus, rectangle, and square. Then copy the properties in Exercises 25–34. After each property, use your symbols to list the quadrilaterals having that property. 25–34. See 25. All sides are . 26. Opposite sides are . back of book. 27. Opposite sides are 6. 28. Opposite ' are . PowerPoint Lesson Quiz 1. The diagonals of a rectangle have lengths 4 + 2x and 6x - 20. Find x and the length of each diagonal. 6; each diagonal has length 16. 1 2 32. Diagonals are . 33. Diagonals are #. 34. Each diagonal bisects opposite '. 37. a quadrilateral that is not a special quadrilateral 38. Writing Summarize the properties of squares that follow from a square being (a) a parallelogram, (b) a rhombus, and (c) a rectangle. See back of book. 70° Proof 2 Proof 4 4. Each diagonal is 15 cm long, and one angle of the quadrilateral has measure 45. Impossible; if diagonals of a parallelogram are congruent, the quadrilateral is a rectangle, but a rectangle has four right angles. C A Given: $ABCD; AC > BD B D Prove: ABCD is a rectangle. See back of book. C Reasoning Decide whether each of these is a good definition. Justify your answer. GO for Help 41. A rectangle is a quadrilateral with four right angles. 41–43. See margin. To review what makes a good definition, see Lesson 2-2. 42. A rhombus is a quadrilateral with four congruent sides. 43. A square is a quadrilateral with four right angles and four congruent sides. x 2 Algebra Find the value(s) of the variable(s) for each parallelogram. 44. RZ = 2x + 5, SW = 5x - 20 R 5. The diagonals are congruent, perpendicular and bisect each other. Yes; if diagonals of a parallelogram are congruent, the quadrilateral is a rectangle, and if diagonals of a parallelogram are perpendicular, the quadrilateral is a rhombus, and a rectangle that is a rhombus is a square. 45. m&1 = 3y - 6 46. BD = 4x - y + 1 GPS B W 9x 1 Z S 2x 1 3y 5 6z A T C 30 x ≠ 5, y ≠ 32, z ≠ 7.5 D x ≠ 7.5, y ≠ 3 Open-Ended Given two segments with lengths a and b (a u b), what special quadrilaterals can you sketch that meet these conditions? Show each sketch. 47–49. See back of book. 47. Both diagonals have length a. 48. The two diagonals have lengths a and b. 49. One diagonal has length a, one side of the quadrilateral has length b. 334 41. Yes; since all right ' are O, the opp. ' are O and it is a ~. Since it has all right ', it is a rectangle. D 3 4 21 B Prove: ABCD is a rhombus. See back of book. 40. Prove Theorem 6-14. ml1 ≠ 90, ml2 ≠ 20, ml3 ≠ 20, ml4 ≠ 70 Determine whether the quadrilateral can be a parallelogram. If not, write impossible. Explain. A 39. Prove Theorem 6-12. Given: ABCD is a parallelogram; AC bisects &BAD and &BCD. 1 334 36. a kite 3 ml1 ≠ 62, ml2 ≠ 62, ml3 ≠ 56 3 31. Diagonals bisect each other. 35. a trapezoid 56° 3. 30. Consecutive ' are supplementary. Which, if any, of the properties in Exercises 25–34 can the following type of quadrilateral have? Draw diagrams to illustrate. 35–37. See back of book. Find the measures of the numbered angles in each rhombus. 2. 29. All ' are right '. Chapter 6 Quadrilaterals 42. Yes; 4 sides are O, so the opp. sides are O making it a ~. Since it has 4 O sides it is also a rhombus. 43. Yes; a quad. with 4 O sides is a ~ and a ~ with 4 O sides and 4 right ' is a square. 60. Impossible; if the diagonals of a ~ are O, then it would have to be a rectangle and have right '. In Exercises 54–59, the “given diagonals” are two segments you draw. The constructed figures must have diagonals that match. 16, 16 50. AC = 2(x - 3) and BD = x + 5 52. AC = 3y 5 and BD = 3y - 4 1, 1 2, 2 51. AC = 2(5a + 1) and BD = 2(a + 1) 1, 1 53. AC = 3c 9 and BD = 4 - c Constructions Explain how to construct each figure, given its diagonals. 54–59. See back of book. 55. rectangle 56. rhombus GPS 54. parallelogram 57. square C Alternative Assessment x 2 Algebra ABCD is a rectangle. Find the length of each diagonal. Problem Solving Hint Challenge 58. kite 59. trapezoid Have students draw theorems without words to illustrate each of the theorems in Lesson 6-4. For example, the figure below illustrates Theorem 6-9: Each diagonal of a rhombus bisects two angles of the rhombus. Determine whether the quadrilateral can be a parallelogram. If not, write impossible. Explain. 60–62. See margin. 60. The diagonals are congruent, but the quadrilateral has no right angles. Test Prep 61. Each diagonal is 3 cm long and two opposite sides are 2 cm long. 62. Two opposite angles are right angles, but the quadrilateral is not a rectangle. Resources Proof 63. In Theorem 6-12, replace “two angles” with “one angle.” Write a paragraph that proves this new statement true or show a counterexample to prove it false. See margin. Test Prep Multiple Choice For additional practice with a variety of test item formats: • Standardized Test Prep, p. 361 • Test-Taking Strategies, p. 356 • Test-Taking Strategies with Transparencies 64. The diagonals of a quadrilateral are perpendicular bisectors of each other. What name best describes the quadrilateral? D A. rectangle B. parallelogram C. quadrilateral D. rhombus 65. The diagonals of a quadrilateral bisect both pairs of opposite angles. What name best describes the quadrilateral? J F. parallelogram G. quadrilateral H. rectangle J. rhombus Short Response 66. Given: QRST is a rhombus, QS intersects RT at P, QR = 9 cm, and QP = 4.5 cm. Find m&RST. Explain your work. See margin. Mixed Review GO for Help Lesson 6-3 Can you conclude that the quadrilateral is a parallelogram? Explain. 5 6 67. 68. 69. B 4 Lesson 5-1 4 25 A 73. SU 6 9 74. TU 6 9 75. PQ 6 9 ST RQ RP Lesson 3-4 x 2 76. Algebra Find the value of c. 89 (c 28) D R 70. TQ = 9 6 71. PQ = 916 72. TU = 9 5 61. Yes; O diagonals in a ~ mean it can be a rectangle with two opposite sides 2 cm long. AB ≠ CD and BC ≠ DA. So AB ≠ BC ≠ CD ≠ DA, and ~ABCD is a rhombus. The new statement is true. 25 5 6 67–69. See margin. In kPQR, points S, T, and U are midpoints. Complete each statement. lesson quiz, PHSchool.com, Web Code: aua-0604 C 5 P S 8 6 U Q c (c 65) Lesson 6-4 Special Parallelograms 62. Impossible; in a ~, consecutive ' must be supp., so all ' must be right. This would make it a rectangle. T 335 63. Given ~ABCD with diagonal AC. Let AC bisect lBAD. Because kABC O kDAC, AB ≠ DA by CPCTC. But since opp. sides of a ~ are O, 66. [2] Since diagonals of a rhombus bisect each other, QS ≠ 9 cm. Also, since all sides are O, RS ≠ 9 cm. So kQRS is an equilateral k and each interior l is 180 3 or 608. kQTS is also an equilateral k, so its ' are 608. By l add. (mlPST ± mlPSR ≠ mlRST), mlRST ≠ 60 ± 60 or 120. [1] no work shown OR a response that is only partially correct 335