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Transcript
7.5 Triangles 7.5 OBJECTIVES 1. 2. 3. 4. 5. Find the measure of the third angle of a triangle Classify triangles as acute, obtuse, or right Classify triangles as equilateral, isosceles, or scalene Recognize similar triangles Apply similar triangles to find an unknown length. Now that you know something about angles, it is interesting to again look at triangles. Why is this shape called a triangle? Literally, triangle means “three angles.” The same classifications we used for angles can be used for triangles. If a triangle has a right angle, we call it a right triangle. If it has three acute angles, it is called an acute triangle. If it has an obtuse angle, it is called an obtuse triangle. Example 1 © 2001 McGraw-Hill Companies Identifying an Acute Triangle Which of the following triangles is acute? E B A C D X F Z Y Only DEF is an acute triangle. Both ABC and XYZ have one obtuse angle. 585 586 CHAPTER 7 GEOMETRY AND MEASURE CHECK YOURSELF 1 Which of the following triangles is obtuse? Z Y B E C A F D X We can also classify triangles based on how many angles have the same measure. A triangle is called an equilateral triangle if all three angles have the same measure. A triangle is called an isosceles triangle if two angles have the same measure. A triangle is called a scalene triangle if no two angles have the same measure. Example 2 Labeling Types of Triangles B E 40 60 Z 60 70 A NOTE Notice that each of these triangles can be classified in different ways. XYZ is a right triangle, but it is also scalene. 60 70 C D 60 30 F Y X ABC is an isosceles triangle because two of the angles have the same measure. And DEF is an equilateral triangle because all three angles have the same measure. XYZ is a scalene triangle because no two angles have the same measure. © 2001 McGraw-Hill Companies Of the following triangles, which are equilateral? Isosceles? Scalene? TRIANGLES SECTION 7.5 587 CHECK YOURSELF 2 Label each triangle as equilateral, isosceles, or scalene. 60 70 80 30 100 60 (1) 60 40 (2) 40 (3) Go back and look at the sum of the angles inside each of the triangles in the last example. You will note that they always add up to 180°. No matter how we draw a triangle, the sum of the three angles inside the triangle will always be 180°. Here is an experiment that might convince you that this is always the case. 1. Using a straight edge, draw any triangle you wish on a sheet of paper. 2. Use scissors to cut out the triangle. 3. Rip the three vertices off of the triangle. 4. Lay the three vertices (with the points of the triangle touching) together. They will always form a straight angle, which we saw in the previous section has a measure of 180°. Rules and Properties: Angles of a Triangle For any triangle ABC, mA mB mC 180° © 2001 McGraw-Hill Companies Example 3 Finding an Angle Measure Find the measure of the third angle in this triangle. ? 53 68 588 CHAPTER 7 GEOMETRY AND MEASURE We need the three measurements to add to 180°, so we add the two given measurements (53° 68° 121°). Then we subtract that from 180° (180° 121° 59°). This gives us the measure of the third angle, 59°. CHECK YOURSELF 3 Find the measure of ABC. B 70 35 C A Definitions: Similar Triangles If the measurements of the three angles in two different triangles are the same, we say the two triangles are similar triangles. Example 4 Identifying Similar Triangles Which two triangles are similar? B E 40 40 Z 70 Y X Although they are of different size, ABC and XYZ are similar because they have the same angle measurements. CHECK YOURSELF 4 Find the two triangles that are similar. E Z B 45 30 60 A C D F Y X © 2001 McGraw-Hill Companies ABC XYZ 40 F C A NOTE This can be written 50 D 70 70 70 TRIANGLES SECTION 7.5 589 Finally, let’s return to an idea that we first saw in Chapter 5. Rules and Properties: Similar Triangles If two triangles are similar, their corresponding sides have the same ratio. The most common use of this property of similar triangles occurs when you wish to find the height of a tall object. Example 5 illustrates this application of similar triangles. Example 5 Finding the Height of a Tree If a man who is 180 cm tall casts a shadow that is 60 cm long, how tall is a tree that casts a shadow that is 9 m long? Note that, because of the sun, the man and his shadow forms a similar triangle to the tree and its shadow. Because of this, we can use the common ratio to find the height of the tree. xm 180 cm © 2001 McGraw-Hill Companies 60 cm 9m 180 cm xm 60 cm 9m x (180 ⋅ 9) 60 m x 27 m The tree is 27 m tall. CHAPTER 7 GEOMETRY AND MEASURE CHECK YOURSELF 5 If a man who is 160 cm tall casts a shadow that is 120 cm long, how tall is a building that casts a shadow that is 60 m long? CHECK YOURSELF ANSWERS 1. XYZ 2. (1) Scalene; (2) equilateral; (3) isosceles 4. DEF and XZY 5. 80 m 3. 75° © 2001 McGraw-Hill Companies 590 Name Exercises 7.5 Section Date In exercises 1 to 4, label the triangles as acute or obtuse. 1. ANSWERS 2. 1. 2. 3. 4. 3. 4. 5. 6. 7. 8. In exercises 5 to 10, label the triangles as equilateral, isosceles, or scalene. 5. 9. 10. 6. 53 60 60 7. 8. 60 40 60 70 © 2001 McGraw-Hill Companies 9. 10. 25 40 120 130 591 ANSWERS 11. In exercises 11 to 16, find the missing angle and then label the triangle as equilateral, isosceles, or scalene. 12. 11. 13. 12. 14. 15. 30 50 120 16. 17. 18. 13. 14. 19. 20. 45 30 15. 16. 67 46 65 50 For each triangle shown, find the indicated angle. 17. Find mC 18. Find mC B C 82 71 A C 19. Find mA 23 B 20. Find mB B B 39 A 18 A 592 C 31 15 C © 2001 McGraw-Hill Companies 61 A ANSWERS 21. 21. Find mB 22. Find mA B 22. C 23. A 24. 18 B 63 A 25. 26. C 27. In exercises 23 and 24, assume that the given triangle is isosceles. 23. Find mA and mC 24. Find mD and mF E B 110 8 72 8 15 A 15 C F D 25. Which two triangles are similar? 60 60 80 50 70 50 (b) (a) (c) 26. Which two triangles are similar? 30 (a) © 2001 McGraw-Hill Companies 25 60 (c) (b) In exercises 27 to 32, the two triangles shown are similar. Find the indicated side. 27. Find v S V 3 12 R 5 T U v W 593 ANSWERS 28. 28. Find f 29. E B 30. 4 6 31. 20 f C A F D 29. Find g K H g 24 I L 14 G 28 J 30. Find m Q N m 10 O M 35 20 R P T W 38.7 50.4 V S 594 t U 30.1 X © 2001 McGraw-Hill Companies 31. Find t ANSWERS 32. 32. Find e 33. D G e E C 34. 28.5 45.6 40.5 H F 35. 36. In exercises 33 to 36, first show that the two triangles are similar. Then find the indicated side. Round your answer to the nearest hundredth. 33. Find KL 37. 34. Find PQ P K ? ? 59 L 70 N M 78 65 Q S R 92 O 98 T 35. Find VX 36. Find AC V A 48 ? 40 ? Y W D X B C 60 82 51 © 2001 McGraw-Hill Companies Z 36 E Find the indicated side. If necessary, round to the nearest tenth of a unit. ___ 37. Find DE D ? B 4 A 8 C 12 E 595 ANSWERS __ 38. 38. Find IJ 39. I 40. ? G 41. 2 42. F H 6 J 9 ___ 39. Find KL L M ? 32 K N 38 O 52 ___ 40. Find PQ Q ? R 14 P S 45 T 32 41. Given: mBCA mDEA ___ Find DE D B ? 17 C 31 E 14 42. Given: __mGHF mIJF Find IJ I ? G 27 F 596 41 H 46 J © 2001 McGraw-Hill Companies A ANSWERS 43. A light pole casts a shadow that measures 4 ft. At the same time, a yardstick casts a shadow that is 9 in. long. How tall is the pole? 43. 44. 45. 46. 47. 44. A tree casts a shadow that measures 5 m. At the same time, a meter stick casts a shadow that is 0.4 m long. How tall is the tree? 45. Use the ideas of similar triangles to determine the height of a pole or tree on your campus. Work with one or two partners. In exercises 46 to 48, one side of the triangle has been extended, forming what is called an “exterior angle.” In each case, find the measure of the indicated exterior angle. 46. 85 © 2001 McGraw-Hill Companies ? 58 47. 44 ? 83 597 ANSWERS 48. 48. 61 ? 98 49. 49. What do you observe from exercises 46 to 48? Write a general conjecture about an 50. exterior angle of a triangle. 51. 50. Write an argument to show that an equilateral triangle cannot have a right angle. 52. 51. Argue that, given an equilateral triangle, the measure of each angle must 53. be 60°. 54. 52. Argue that a triangle cannot have more than one obtuse angle. 53. Is it possible to have an isosceles right triangle? If such a triangle exists, what can be said about the angles? Defend your statements. If the sum of the measures of two angles is 90°, the two angles are said to be complementary. If A and B are complementary angles, A is said to be the complement of B, and B is said to be the complement of A. 54. Create an argument to support the following statement: If ABC is a right triangle, with mC 90°, then A and B must be acute and complementary. 1. Acute 3. Acute 5. Equilateral 7. Isosceles 9. Isosceles 11. 30°; isosceles 13. 45°; isosceles 15. 67°; isosceles 17. 37° 19. 123° 21. 27° 23. A 35°; C 35° 25. b and c 27. 20 29. 12 31. 39.2 33. 77.12 35. 65.60 37. 10 39. 55.4 41. 24.7 43. 16 ft 45. 47. 127° 49. 51. 598 53. © 2001 McGraw-Hill Companies Answers