9.3. Practice C For use with pages Tell whether the triangle is a right triangle.

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1 LESSON 9.3 NAME DATE For use with pages Tell whether the triangle is a right triangle Decide whether the numbers can represent the side lengths of a triangle. If the can, classif the triangle as right, acute, or obtuse. 7. 7, 5, 3 6., 12, , 10. 7, 11, , 7, , 72, , 5 2 Lesson 9.3 Classif the quadrilateral. Eplain how ou can prove that the quadrilateral is that tpe Quadrilateral QUAD has vertices at Q 5, 2, U 1, 7, A 4, 3, and D 0, 2. Plot the figure and indicate what tpe of quadrilateral QUAD is. Find the perimeter of QUAD Write a two-column proof or a paragraph proof. 17. Given: AB 6, BC, AC Given: XZ 3, ZY 6, KY Prove: 1 is a right angle Prove: 2 is obtuse B Z 6 X Y A Geometr Chapter 9 Resource Book C Copright McDougal Littell Inc. All rights reserved.

2 Answer Ke Chapter 9 Lesson es 2. es 3. no 4. no 5. no 6. es 7. es, right. es, obtuse 9. es, acute 10. es, obtuse 11. es, right 12. es, right 13. Kite; so b the Converse of the Pthagorean Thm. the diagonals are, also two pairs of consecutive sides are congruent (use congruent triangles), thus the quad. is a kite. 14. Square; the diagonals bisect each other so the quad. is a parallelogram; the diagonals are congruent so the parallelogram is a rectangle; so the diagonals intersect at a right angle to form lines; thus the parallelogram is also a rhombus. A quad. that is both a rectangle and a rhombus is a square. 15. Rectangle; opposite sides are congruent therefore it is a parallelogram; so one pair of opposite angles are right angles; it follows that the parallelogram is a rectangle. 16. U Quad QUAD is a square; perimeter 4 41 Q 1 1 A D 17. Since , ABC is a right b the Converse of the Pthagorean Thm. ABC is a right since it is opposite the longest side. ABC and 1 are a linear pair b def. So Therefore, b def. 2 is a right. 1. Since < 2, XYZ is an obtuse b Thm XYZ is obtuse since it is opposite the longest side. Since vertical angles are congruent, XYZ 2. So b substitution, 2 is obtuse.

3 LESSON 9.4 NAME DATE For use with pages Find the value of each variable. Write answers in simplest radical form Sketch the figure that is described. Find the requested length, perimeter, or area. Round decimals to the nearest tenth. 10. The altitude of an equilateral triangle is 12 centimeters. Find the perimeter of the triangle. 11. The diagonal of a square is inches. Find the area. 12. The perimeter of a rectangle is 66 centimeters. The length is twice the width. Find the length of the diagonal. 13. The perimeter of an equilateral triangle is 36 inches. Find the length of an altitude. 14. Each figure below is a Each figure below is a triangle. triangle. Find the value of. Find the value of Lesson 9.4 Copright McDougal Littell Inc. All rights reserved. Geometr 59 Chapter 9 Resource Book

4 Answer Ke Chapter 9 Lesson , , , 9 4., , , , 15., , perimeter 41.6 cm 11. area 32 in cm in diagonal 24.6 cm altitude 10.4 inches 12 in. 12 in in.

5 Lesson 9.5 LESSON 9.5 NAME DATE For use with pages Find the sine, the cosine, and the tangent of the acute angles of the triangle. Epress each answer as a decimal rounded to four places. 1. M 2. R T 6 6 A 20 A C B 7 J 6 2 E Use a calculator to approimate the given value to four decimal places. 4. sin cos 3 6. tan tan cos sin sin 71 cos 64 Find the value of each variable. Round decimals to the nearest tenth In Eercises 15 17, use the figure of the lighthouse. 15. At 2 P.M. the shadow of a lighthouse is 22 feet long and the angle of elevation is 72. Find the height of the lighthouse. 16. At 4 P.M. the angle of elevation of the sun is 40. Find the length of the shadow cast b the lighthouse. 17. At 6 P.M. will the length of the shadow be longer or shorter than it was at 4 P.M.? Eplain ft In Eercises 1 and 19, use the figure of the escalator. 1. A new store is being built. An escalator is planned. It will make an angle of 34 with the floor. If the vertical distance between floors is 14 feet, how long will the escalator be? 19. If the angle made with the floor is changed to 36, will the length of the escalator increase or decrease? Eplain ft 74 Geometr Chapter 9 Resource Book Copright McDougal Littell Inc. All rights reserved.

6 Answer Ke Chapter 9 Lesson sin T , cos T 0.321, tan T , sin A 0.321, cos A , tan A sin J cos J sin E cos E , tan J tan E 1 3. sin A 0.2, cos A 0.96, tan A , sin B 0.96, cos B 0.2, tan B , , , about 67.7 ft 16. about 0.7 ft 17. Longer; as the sun sets the angle decreases and the tangent of the angle also decreases. The height of the lighthouse is constant so the shadow has to lengthen for the ratio to get smaller. 1. about 25.0 ft 19. Shortens b about 1.2 feet; if the angle increases, so does the sine of the angle. The vertical distance between floors is constant so the length of the escalator must shorten for the ratio to increase.

7 Lesson 9.5 LESSON 9.5 NAME DATE Challenge: Skills and Applications For use with pages Refer to the diagram. What is the length of the base Round to the nearest tenth. AB? C 1 A 55 D 65 B 2. When the sun is shining at a 62 angle of elevation, a flagpole forms a shadow of length feet. Later, the sun shines at an angle of 40, and the shadow is 25 feet longer than before. a. Write two epressions for the height DE of the flagpole, in terms of. b. How tall is the flagpole? Round to the nearest tenth of a foot. 3. Refer to the diagram. Find JK and KL. Round decimals to the nearest tenth. (Hint: Draw an altitude.) J D E 62 F 40 G ft 25 ft In Eercises 4 6, refer to the diagram. 4. Write an epression for tan and an epression for tan 90, in terms of a, b, and c. How is the tangent of an angle related to the tangent of the angle s complement? 5. Write an epression for sin 2 cos 2 in terms of a, b, and c. Then use the Pthagorean Theorem to simplif our epression. 6. If sin 0.6, what is the value of cos? 7. Complete the following steps to evaluate sin 1. a. Show that PRQ ~ QRT. b. Use the similar triangles to find and solve a proportion involving. c. Name an 1 angle in the diagram. What is the eact value of sin 1, in radical form? 42 K a c b P 36 1 T 1 S L R 1 Q 0 Geometr Chapter 9 Resource Book Copright McDougal Littell Inc. All rights reserved.

8 Answer Ke Chapter 9 Lesson 9.5 Challenge: Skills and Applications a. tan 62, 25 tan 40 b ft 3. JK 40.1, KL tan a the are b, tan 90 b a ; reciprocals. 5. a 2 b 2 c 2 sin 2 cos 2 a c2 c a. Sample answer: Since PT QT, we have PQT P, so m PQT 36. So b the Eterior Angle Theorem, m QTR m P m PQT Appling the Base Angles Theorem again, since QT QR, we have m R m QTR 72, so m RQT So RQT RPQ, and R R (Refleive Propert of Congruence); therefore, PRQ ~ QRT (AA Similarit Postulate) ± 5 b. ; c. SQT or RQS ; 4 c 2 b c 2

9 LESSON 9.6 NAME DATE For use with pages In Eercises 1, A is an acute angle. Use a calculator to approimate the measure of A. Round to one decimal place. 1. sin A tan A cos A cos A tan A sin A sin A 0.06 tan A 0.4 Solve the right triangle. Round decimals to the nearest tenth. Lesson A 10. J H 53 C 16 B M O 13 I 27 P 12. S 13. X Z Q T T 32 R Y R 15. D E 16. M 17. N C 56 T F L.2 N 1. Submarine A sonar operator on a ship detects a submarine at a distance of 400 meters and an angle of depression of 35. How deep is the submarine? m 19. Height of a Building Two buildings are 60 feet apart across a street. A person on top of the shorter building finds the angle of elevation of the roof of the taller building to be 20 and the angle of depression of its base to be 35. How tall is the taller building to the nearest foot? sidewalk 90 Geometr Chapter 9 Resource Book Copright McDougal Littell Inc. All rights reserved.

10 Answer Ke Chapter 9 Lesson AB 17.9, m A 63.4, m B 26.6 m J 37, IJ 1.4, HJ 13. m M 63, MO 6.6, MP 14.6 ST 17.2, m T 57.5, m R 32.5 m Z 62.5, XY 15., XZ 17. RT 11.9, m R 51.5, m T 3.5 m D 59, DE 10.5, DF 20.4 MN 25.1, m N 70.9, m M 19.1 m T 34, NC 10.4, CT about m 19. about 63. ft

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