11.4 AA Similarity of Triangles

Transcription

1 Name lass ate 11.4 Similarity of Triangles ssential Question: How can you show that two triangles are similar? xplore G.7. pply the ngle-ngle criterion to verify similar triangles and apply the proportionality of the corresponding sides to solve problems. lso G.5. xploring ngle-ngle Similarity for Triangles esource Locker Two triangles are similar when their corresponding sides are proportional and their corresponding angles are congruent. There are several shortcuts for proving triangles are similar. raw a triangle and label it. lsewhere on your page, draw a segment longer than _ and label the endpoints and. opy and to points and, respectively. xtend the rays of your copied angles, if necessary, and label their intersection point. You have constructed. Houghton Mifflin Harcourt Publishing ompany You constructed angles and to be congruent to angles and, respectively. Therefore, angles and must also be because of the Theorem. heck the proportionality of the corresponding sides. _ = _ = _ = _ = _ = _ = Since the ratios are the sides of the triangles are. eflect 1. iscussion ompare your results with your classmates. What conjecture can you make about two triangles that have two corresponding congruent angles? Module Lesson 4

2 xplain 1 Proving ngle-ngle Triangle Similarity The xplore suggests the following theorem for determining whether two triangles are similar. ngle-ngle () Triangle Similarity Theorem If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. xample 1 Prove the ngle-ngle Triangle Similarity Theorem. Given: X and Y Prove: XYZ Y X Z pply a dilation to with scale factor k = _ XY. Let the image of be. dilation Y X Z is similar to, and and because. lso, = k =. It is given that X and Y y the Transitive Property of ongruence, and. So, XYZ by. This means there is a sequence of rigid motions that maps to YYZ. Houghton Mifflin Harcourt Publishing ompany The dilation followed by this sequence of rigid motions shows that there is a sequence of similarity transformations that maps to XYZ. Therefore, XYZ. Module Lesson 4

3 eflect 2. iscussion ompare and contrast the Similarity Theorem with the S ongruence Theorem. 3. In JKL, m J = 40 and m K = 55. In MNP, m M = 40 and m P = 5. student concludes that the triangles are not similar. o you agree or disagree? Why? xplain 2 pplying ngle-ngle Similarity rchitects and contractors use the properties of similar figures to find any unknown dimensions, like the proper height of a triangular roof. They can use a bevel angle tool to check that the angles of construction are congruent to the angles in their plans. xample 2 ind the indicated length, if possible. Houghton Mifflin Harcourt Publishing ompany Image redits: John Lund/rew Kelly/lend Images/orbis irst, determine whether. y the lternate Interior ngles Theorem, and, so by the Triangle Similarity Theorem. ind by solving a proportion. _ = _ 54_ 36 = _ _ = _ = Module Lesson 4

4 T S T V U 12 heck whether SV TU: It is given in the diagram that. is shared by both triangles, so by the Property of ongruence. So, by the, ST TU. ind T by solving a proportion. _ T S = TU_ SV T_ = _ T = eflect 4. In xample 2, is there another way you can set up the proportion to solve for? 5. iscussion When asked to solve for y, a student sets up the proportion as shown. xplain why the proportion is wrong. How should you adjust the proportion so that it will give the correct result? _ y = _ 14 y 14 Houghton Mifflin Harcourt Publishing ompany Module Lesson 4

5 Your Turn 6. builder was given a design plan for a triangular roof as shown. xplain how he knows that. Then find. 9 ft. 15 ft. 6 ft. 7. ind PQ, if possible. P Q T 9 15 S 12 xplain 3 pplying SSS and SS Triangle Similarity In addition to ngle-ngle Triangle Similarity, there are two additional shortcuts for proving two triangles are similar. Side-Side-Side (SSS) Triangle Similarity Theorem If the three sides of one triangle are proportional to the corresponding sides of another triangle, then the triangles are similar. Side-ngle-Side (SS) Triangle Similarity Theorem If two sides of one triangle are proportional to the corresponding sides of another triangle and their included angles are congruent, then the triangles are similar. Houghton Mifflin Harcourt Publishing ompany xample 3 etermine whether the given triangles are similar. Justify your answer. M 4 6 N You are given two pairs of corresponding side lengths and one pair of congruent corresponding angles, so try using SS. P 4 Q heck that the ratios of corresponding sides are equal. _ MN M = 4_ 6 = 2_ 3 _ MP MQ = _ + 4 = _ 12 = 2_ 3 heck that the included angles are congruent: NMP QM is given in the diagram. Therefore NMP MQ by the SS Triangle Similarity Theorem. Module Lesson 4

6 M 4 J G H 12 L 15 6 N You are given pairs of corresponding side lengths and congruent corresponding angles, so try using. heck that the ratios of corresponding sides are equal. LM_ GH = _ = _ _ MN = _ = _ _ = _ = _ Therefore by. Since you are given all three pairs of sides, you don t need to check for congruent angles. eflect. re all isosceles right triangles similar? xplain why or why not. 9. Why isn't ngle-side-ngle (S) used to prove two triangles similar? Your Turn If possible, determine whether the given triangles are similar. Justify.. H G Houghton Mifflin Harcourt Publishing ompany Module Lesson 4

7 11. M 5 J 3 N O G H laborate 12. Is triangle similarity transitive? If you know and GHJ, is GHJ? xplain. 13. The Similarity Theorem applies to triangles. Is there an Similarity Theorem for quadrilaterals? Use your geometry software to test your conjecture or create a counterexample. Houghton Mifflin Harcourt Publishing ompany 14. ssential Question heck-in How can you prove triangles are similar? Module Lesson 4

8 valuate: Homework and Practice Show that the triangles are similar by measuring the lengths of their sides and comparing the ratios of the corresponding sides. Online Homework Hints and Help xtra Practice 1. _ = _ = _ = _ = _ = _ = 2. _ = _ = _ = _ = _ = _ = etermine whether the two triangles are similar. If they are similar, write the similarity statement Houghton Mifflin Harcourt Publishing ompany Module Lesson 4

9 Houghton Mifflin Harcourt Publishing ompany xplain how you know whether the triangles are similar. If possible, find the indicated length Module Lesson 4

10 11. Q 12. ind. P Q ST P S.0 Q T Show whether or not each pair of triangles is similar. Justify your answer, and write a similarity statement when the triangles are similar Q P Houghton Mifflin Harcourt Publishing ompany Module Lesson 4

11 H 20. H G J P. 60 Q G J P 60 Q lgebra ind all possible values of x for which these two triangles are similar. (x + ) Houghton Mifflin Harcourt Publishing ompany x (2x - 50) 22. xplain the rror student analyzes the two triangles shown below. xplain the error that the student makes = 5.5 = 1.15, and 4. = 6 6 = 1 ecause the two ratios are not equal, the two triangles are not similar. Module Lesson 4

12 23. Multi-Step Identify two similar triangles in the figure, and explain why they are similar. Then find The picture shows a person taking a pinhole photograph of himself. Light entering the opening reflects his image on the wall, forming similar triangles. What is the height of the image to the nearest inch? 15 in. 4 ft 6 in. 5 ft 5 in. H.O.T. ocus on Higher Order Thinking 25. nalyze elationships Prove the SS Triangle Similarity Theorem. Y X Z Given: _ XY = _ XZ and X Prove: XYZ Houghton Mifflin Harcourt Publishing ompany Module Lesson 4

13 26. nalyze elationships Prove the SSS Triangle Similarity Theorem. Y X Z Given: _ XY = _ XZ = _ YZ Prove: XYZ (Hint: The main steps of the proof are similar to those of the proof of the Triangle Similarity Theorem.) 27. ommunicate Mathematical Ideas student is asked to find point X on such that X and X is as small as possible. The student does so by constructing a perpendicular line to at point, and then labeling X as the intersection of the perpendicular line with. xplain why this procedure generates the similar triangle that the student was requested to construct. Houghton Mifflin Harcourt Publishing ompany X 6 Module Lesson 4

14 2. Make a onjecture uilders and architects use scale models to help them design and build new buildings. n architecture student builds a model of an office building in which the height of the model is 1 of the height of the actual building, while 400 the width and length of the model are each 1 of the corresponding dimensions of 200 the actual building. The model includes several triangles. escribe how a triangle in this model could be similar to the corresponding triangle in the actual building, then describe how a triangle in this model might not be similar to the corresponding triangle in the actual building. Use a similarity theorem to support each answer. Lesson Performance Task The figure shows a camera obscura and the object being photographed. nswer the following questions about the figure: 1. xplain how the image of the object would be affected if the camera were moved closer to the object. How would that limit the height of objects that could be photographed? G 2. How do you know that is similar to G? 3. Write a proportion you could use to find the height of the pine tree. 4. = 12 in., G = in., = 96 ft. How tall is the pine tree? Houghton Mifflin Harcourt Publishing ompany Module Lesson 4