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12.4 Using Similar Triangles How can you use angles to tell whether triangles are similar? 1 ACTIVITY: Constructing Similar Triangles Use a straightedge to draw a line segment that is 4 centimeters long. ● Then use the line segment and a protractor to draw a triangle that has a 60° and a 40° angle, as shown. Label the triangle ABC. 80 90 10 0 70 10 0 90 80 110 1 70 20 60 0 110 60 1 2 3 50 0 1 50 0 3 1 60í 170 180 60 0 1 20 10 0 15 0 30 14 0 4 ● 0 10 180 170 1 20 3 60 15 0 4 01 0 40 Work with a partner. 40í 4 cm a. Explain how to draw a larger triangle that has the same two angle measures. Label the triangle JKL. b. Explain how to draw a smaller triangle that has the same two angle measures. Label the triangle PQR. c. Are all of the triangles similar? Explain. 2 ACTIVITY: Using Technology to Explore Triangles Work with a partner. Use geometry software to draw the triangle below. A 50î Geometry 30î B In this lesson, you will ● understand the concept of similar triangles. ● identify similar triangles. ● use indirect measurement to find missing measures. C a. Dilate the triangle by the following scale factors. 2 1 2 — 1 4 — 2.5 b. Measure the third angle in each triangle. What do you notice? c. REASONING You have two triangles. Two angles in the first triangle are congruent to two angles in the second triangle. Can you conclude that the triangles are similar? Explain. 550 Chapter 12 ms_accel_pe_1204.indd 550 Angles and Triangles 2/24/15 8:10:54 AM 3 Math Practice Make Sense of Quantities What do you know about the sides of the triangles when the triangles are similar? ACTIVITY: Indirect Measurement Work with a partner. F a. Use the fact that two rays from the Sun are parallel to explain why △ABC and △DEF are similar. b. Explain how to use similar triangles to find the height of the flagpole. x ft Sun’s ray C Sun’s ray 5 ft A 3 ft B D 36 ft E 4. IN YOUR OWN WORDS How can you use angles to tell whether triangles are similar? 5. PROJECT Work with a partner or in a small group. a. Explain why the process in Activity 3 is called “indirect” measurement. b. CHOOSE TOOLS Use indirect measurement to measure the height of something outside your school (a tree, a building, a flagpole). Before going outside, decide what materials you need to take with you. c. MODELING Draw a diagram of the indirect measurement process you used. In the diagram, label the lengths that you actually measured and also the lengths that you calculated. 6. PRECISION Look back at Exercise 17 in Section 11.5. Explain how you can show that the two triangles are similar. Use what you learned about similar triangles to complete Exercises 4 and 5 on page 554. Section 12.4 ms_accel_pe_1204.indd 551 Using Similar Triangles 551 2/24/15 8:11:03 AM 12.4 Lesson Lesson Tutorials Key Vocabulary indirect measurement, p. 553 Angles of Similar Triangles Words When two angles in one triangle are congruent to two angles in another triangle, the third angles are also congruent and the triangles are similar. Example E B 95í 95í 65í 65í 20í A C 20í D F Triangle ABC is similar to Triangle DEF : △ABC ∼ △DEF . EXAMPLE Identifying Similar Triangles 1 Tell whether the triangles are similar. Explain. a. 75í The triangles have two pairs of congruent angles. yí xí 50í 75í 50í So, the third angles are congruent, and the triangles are similar. Write and solve an equation to find x. b. 54í x + 54 + 63 = 180 x + 117 = 180 xí 63í 63í x = 63 yí 63í The triangles have two pairs of congruent angles. So, the third angles are congruent, and the triangles are similar. Write and solve an equation to find x. c. yí xí x + 90 + 42 = 180 x + 132 = 180 42í x = 48 38í The triangles do not have two pairs of congruent angles. So, the triangles are not similar. 552 Chapter 12 ms_accel_pe_1204.indd 552 Angles and Triangles 2/24/15 8:11:12 AM Tell whether the triangles are similar. Explain. 1. Exercises 6–9 2. 80î 28î 66í xî xí yî 28î 24í 71î yí Indirect measurement uses similar figures to find a missing measure when it is difficult to find directly. EXAMPLE A 60 ft B 2 Using Indirect Measurement You plan to cross a river and want to know how far it is to the other side. You take measurements on your side of the river and make the drawing shown. (a) Explain why △ABC and △DEC are similar. (b) What is the distance x across the river? a. ∠ B and ∠ E are right angles, so they are congruent. ∠ ACB and ∠ DCE are vertical angles, so they are congruent. 50 ft C 40 ft E Because two angles in △ABC are congruent to two angles in △DEC, the third angles are also congruent and the triangles are similar. x D b. The ratios of the corresponding side lengths in similar triangles are equal. Write and solve a proportion to find x. x 60 40 50 —=— ⋅ 60x Write a proportion. ⋅ 40 50 60 — = 60 — x = 48 Multiplication Property of Equality Simplify. So, the distance across the river is 48 feet. Exercise 13 3. WHAT IF? The distance from vertex A to vertex B is 55 feet. What is the distance across the river? Section 12.4 ms_accel_pe_1204.indd 553 Using Similar Triangles 553 2/24/15 8:11:17 AM 12.4 Exercises Help with Homework 1. REASONING How can you use similar triangles to find a missing measurement? 2. WHICH ONE DOESN’T BELONG? Which triangle does not belong with the other three? Explain your reasoning. E B K 82î 85í 35î 82í 63î A 82í H C 32í J 35í 63í L 35í 63í G 63í D I F 3. WRITING Two triangles have two pairs of congruent angles. In your own words, explain why you do not need to find the measures of the third pair of angles to determine that they are congruent. 6)=3 9+(- 3)= 3+(- 9)= 4+(- = 1) 9+(- Make a triangle that is larger or smaller than the one given and has the same angle measures. Find the ratios of the corresponding side lengths. 4. 5. 60í 100í 20í 30í 60í Tell whether the triangles are similar. Explain. 6. 39î 7. 34î xî yî 36í 34î 9. 85í 26í 64í 33í xí 72í 39î 8. 81í yí 85í yí 72í 51í 48í xí xí CM 2 3 4 5 6 7 8 9 10 11 12 xí 13 60í 3 2 1 1 yí 48í INCH 1 1 1 4 6 2 4 5 6 2 5 3 8 3 7 5 6 11 10 4 9 4 xxíí 13 12 5 3 45í 6 2 10. RULERS Which of the rulers are similar in shape? Explain. 1 2 3 4 5 6 7 8 9 10 11 ms_accel_pe_1204.indd 554 1 Chapter 12 INCH 554 Angles and Triangles 2/24/15 8:11:24 AM Tell whether the triangles are similar. Explain. 12. 11. 51í 29í 88í 102í 91í 2 13. TREASURE The map shows the number of steps you must take to get to the treasure. However, the map is old, and the last dimension is unreadable. Explain why the triangles are similar. How many steps do you take from the pyramids to the treasure? 240 steps 80 steps 300 steps 14. CRITICAL THINKING The side lengths of a triangle are increased by 50% to make a similar triangle. Does the area increase by 50% as well? Explain. 15. PINE TREE A person who is 6 feet tall casts a 3-foot-long shadow. A nearby pine tree casts a 15-foot-long shadow. What is the height h of the pine tree? 16. OPEN-ENDED You place a mirror on the ground 6 feet from the lamppost. You move back 3 feet and see the top of the lamppost in the mirror. What is the height of the lamppost? 17. REASONING In each of two right triangles, one angle measure is two times another angle measure. Are the triangles similar? Explain your reasoning. D C B 18. In the diagram, segments BG, CF, and DE are parallel. The length of segment BD is 6.32 feet, and the length of segment DE is 6 feet. Name all pairs of similar triangles in the diagram. Then find the lengths of segments BG and CF. A G F E Solve the equation for y. (Topic 3) 19. y − 5x = 3 20. 4x + 6y = 12 1 4 21. 2x − — y = 1 22. MULTIPLE CHOICE What is the value of x? (Section 12.2) A 17 ○ B 62 ○ C 118 ○ D 152 ○ (x à 16)í 40í Section 12.4 ms_accel_pe_1204.indd 555 xí Using Similar Triangles 555 2/24/15 8:11:30 AM