11.4 AA Similarity of Triangles

Transcription

1 Locker LSSON 11.4 Similarity of Triangles ommon ore Math Standards The student is expected to: G-SRT. Use the properties of similarity transformations to establish the criterion for two triangles to be similar. lso G-SRT.5 Mathematical Practices MP.5 Using Tools Name lass ate 11.4 Similarity of Triangles ssential Question: How can you show that two triangles are similar? xplore xploring ngle-ngle Similarity for Triangles 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. Resource Locker Language Objective xplain to a partner how to use the ngle-ngle criterion to show similarity in triangles. NGG ssential Question: How can you show that two triangles are similar? We can use the, SSS, or SS similarity criteria to prove that triangles are similar. PRVIW: LSSON PRORMN TSK View the ngage section online. iscuss the illustration and ask students to speculate on what it depicts. Then preview the Lesson Performance Task. Houghton Mifflin Harcourt Publishing ompany opy and to points and, respectively. xtend the rays of your copied angles, if necessary, and label their intersection point. You have constructed. You constructed angles and to be congruent to angles and, respectively. Therefore, angles and must also be congruent because of the Third ngle Theorem. heck the proportionality of the corresponding sides. Possible answer (ratios should be equal): Since the ratios are equal, the sides of the triangles are proportional. Reflect 1. iscussion ompare your results with your classmates. What conjecture can you make about two triangles that have two corresponding congruent angles? If two triangles have two corresponding congruent angles, the triangles must be similar. Module Lesson 4 Name lass ate 11.4 Similarity of Triangles ssential Question: How can you show that two triangles are similar? G-SRT. Use the properties of similarity transformations to establish the criterion for two triangles to be similar. lso G-SRT.5 xplore xploring ngle-ngle Similarity Houghton Mifflin Harcourt Publishing ompany for Triangles 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. 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. Reflect Resource congruent Third ngle Possible answer (ratios should be equal): equal, proportional 1. iscussion ompare your results with your classmates. What conjecture can you make about two triangles that have two corresponding congruent angles? If two triangles have two corresponding congruent angles, the triangles must be similar. Module Lesson 4 HROVR Turn to Lesson 11.4 in the hardcover edition. 611 Lesson 11.4

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. XPLOR xploring ngle-ngle Similarity for Triangles xample 1 Prove the ngle-ngle Triangle Similarity Theorem. Given: X and Y Prove: XYZ dilation pply a dilation to with scale factor k _ XY. Let the image of be. is similar to, and and because corresponding angles of similar triangles are congruent. XY_ XY lso, k. It is given that X and Y y the Transitive Property of ongruence, X and Y. So, XYZ by the S Triangle ongruence Theorem. This means there is a sequence of rigid motions that maps to YYZ. The dilation followed by this sequence of rigid motions shows that there is a sequence of similarity transformations that maps to XYZ. Therefore, XYZ. Reflect. iscussion ompare and contrast the Similarity Theorem with the S ongruence Theorem. oth theorems require that two pairs of angles be congruent, but for congruence you also X need to know that the included sides are congruent, so that the figures are the same size. The Similarity Theorem shows only that the two triangles are the same shape. Module Lesson 4 PROSSIONL VLOPMNT Learning Progression Students have already proved triangles congruent using SS, SSS, and S. The same kind of reasoning is used here to explore similarity, and to use this similarity to solve problems. X Y Z Y Z Houghton Mifflin Harcourt Publishing ompany INTGRT THNOLOGY Students have the option of doing the similar triangles activity either in the book or online. QUSTIONING STRTGIS What does ngle-ngle similarity claim about triangles? ccording to the ngle-ngle similarity criterion, triangles with two pairs of congruent angles are similar. XPLIN 1 Proving ngle-ngle Triangle Similarity INTGRT MTHMTIL PRTIS ocus on Math onnections MP.1 ompare proving two triangles similar with proving two triangles congruent. QUSTIONING STRTGIS How do you use similarity to show two triangles are similar? Show that two angles of one triangle are congruent to two angles of the other triangle. This lets you conclude that the two triangles are similar. Similarity of Triangles 61

3 XPLIN pplying ngle-ngle Similarity INTGRT MTHMTIL PRTIS ocus on ommunication MP. Remind students that, in triangle similarity, they should identify sides that are proportional, rather than congruent.. 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? isagree; by the Triangle Sum Theorem, m N 55, so the triangles are similar by the xplain 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 Similarity Theorem. ind the indicated length, if possible. irst, determine whether. Houghton Mifflin Harcourt Publishing ompany Image redits: John Lund/rew Kelly/lend Images/orbis RT y the lternate Interior ngles Theorem, and, so by the Triangle Similarity Theorem. ind by solving a proportion. _ 54 6 _ 54 _ _ heck whether RSV RTU: It is given in the diagram that RSV T. R is shared by both triangles, Reflexive so R R by the Property of ongruence. Triangle Similarity Theorem So, by the, RST RTU. ind RT by solving a proportion. _ RT RS TU_ SV 1 RT 6 54 R V 54 S U T 1 RT 15 Module Lesson 4 OLLORTIV LRNING Small Group ctivity Have students work in small groups and draw diagrams to illustrate each of these statements: all squares are similar; all rectangles are not similar; if two polygons are congruent, they are also similar; all right triangles are not similar. 61 Lesson 11.4

4 Reflect 4. In xample _, _ is there another way you can set up the proportion to solve for? Yes; would also give the correct result 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 _ y _ 14 Your Turn 14 9 ft. 15 ft. 6 ft. 6. builder was given a design plan for a triangular roof as shown. xplain how he knows that. Then find. 7. ind PQ, if possible. The variable y does not refer to the side of a triangle but just a segment of a side. y + 14 would be a correct way to solve for y. y the orresponding ngles Theorem, and (or by the Reflexive Property of ongruence), so by the Triangle Similarity Theorem. 6 9 feet 15 QUSTIONING STRTGIS How can you use the similarity postulate to find unknown dimensions? You can use similarity to establish that two triangles with two congruent pairs of angles are similar, and then write a proportion to find unknown lengths of sides. VOI OMMON RRORS Some students may use an incorrect sequence of points when writing a similarity statement. ompare the process to writing a congruence statement and remind them to list corresponding vertices in the same order. T P Q 9 15 S 1 xplain R 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 TSQ is a right angle because PS _ and TR _ are perpendicular. So PRS TQS by the Triangle Similarity Theorem. Let x PQ. _ x + 9 _ x 11 So, PQ 11. If the three sides of one triangle are proportional to the corresponding sides of another triangle, then the triangles are similar. Houghton Mifflin Harcourt Publishing ompany 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. Module Lesson 4 IRNTIT INSTRUTION ognitive Strategies Have students write their own Similarity Theorems. Then ask them to explain why this theorem is not necessary. n Similarity Theorem is not required because of this theorem: Given two triangles, if two pairs of corresponding angles are congruent, then the remaining pair of corresponding angles must also be congruent. Similarity of Triangles 614

5 XPLIN pplying SSS and SS for Triangle Similarity xample etermine whether the given triangles are similar. Justify your answer. N 4 4 M Q P 6 R INTGRT MTHMTIL PRTIS ocus on Math onnections MP.1 Remind students that similarity statements indicate corresponding parts in the same way congruence statements do. You are given two pairs of corresponding side lengths and one pair of congruent corresponding angles, so try using SS. heck that the ratios of corresponding sides are equal. _ MN MR 4_ 6 MP MQ _ + 4 _ 1 _ heck that the included angles are congruent: NMP QMR is given in the diagram. Therefore NMP RMQ by the SS Triangle Similarity Theorem. M J 4 G H 1 L 15 6 N You are given three pairs of corresponding side lengths and zero congruent corresponding angles, so try using the SSS Triangle Similarity Theorem. heck that the ratios of corresponding sides are equal. Houghton Mifflin Harcourt Publishing ompany LM_ GH 1 _ MN 15 _ GJ 6 HJ LN 4 Therefore GHJ LMN by the SSS Triangle Similarity Theorem. Since you are given all three pairs of sides, you don t need to check for congruent angles. Reflect. re all isosceles right triangles similar? xplain why or why not. Let the legs of one isoc. triangle be x and the legs of another be y. Then the ratios of the sides would be x_ y and x_ y. The included angle in each triangle is the right angle, so two isosceles right triangles are similar by SS Similarity. 9. Why isn't ngle-side-ngle (S) used to prove two triangles similar? S implies two pairs of angles are congruent, which is sufficient to show the triangles similar, so checking the included sides is unnecessary. Module Lesson 4 LNGUG SUPPORT onnect Vocabulary Relate the idea of proof to justifying ideas in math. You use established rules and conventions to draw some conclusions. In real life, proof means showing something by gathering evidence by established rules and conventions. 615 Lesson 11.4

6 Your Turn If possible, determine whether the given triangles are similar. Justify your answer.. H The two triangles cannot be proven similar. lthough 5 the two given sides are in proportion, there is not a 6 G pair of included congruent angles. QUSTIONING STRTGIS Why are S and S not similarity theorems? oth of these contain two pairs of corresponding congruent angles, so S and S triangles are already similar by the similarity theorem. 11. M N J 5 O G H y the Pythagorean Theorem, NO 6 and GH 4, so HJ NO GH MN GJ _ MO 1_ SSS Triangle Similarity Theorem.. MNO GHI by the VOI OMMON RRORS Some students may have difficulty identifying corresponding sides in similar triangles because of the orientation of the figures. Show these students how they can copy one of the triangles onto a piece of paper, then cut it out and rotate it, so that the two triangles have the same orientation. laborate 1. Is triangle similarity transitive? If you know and GHJ, is GHJ? xplain. Yes. If the first two triangles have three pairs of congruent angles and the second two triangles do as well, then the first and third triangles will also have those three pairs of congruent angles. 1. 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. No; a square and a rectangle each have three pairs of right angles, but they won t be similar because the sides aren t proportional. 14. ssential Question heck-in How can you prove triangles are similar? Triangles are similar when their corresponding angles are congruent and their corresponding sides are in proportion, but it is sufficient to use Similarity (show two pairs of congruent angles), SSS Similarity (show three pairs of sides in proportion), or SS Similarity (show two pairs of sides are in proportion and their included angles are congruent). Module Lesson 4 Houghton Mifflin Harcourt Publishing ompany LORT QUSTIONING STRTGIS Two isosceles triangles have congruent vertex angles. xplain why the two triangles must be similar. Let the measure of the vertex angles be x. Then, by the Isosceles Triangle Theorem, the base angle in each of the triangles must measure half of (10 - x). So, the triangles are similar by similarity. SUMMRIZ TH LSSON Which theorems allow you to conclude that triangles are similar without using transformations to map one to the other? What do you need to know before you can apply them? The similarity Theorem, the SSS similarity Theorem, and the SS similarity Theorem; you need to know than the triangles have two pairs of congruent angles, or that all three pairs of sides are proportional, or that two pairs of sides are proportional and the included angles are congruent. Similarity of Triangles 616

7 VLUT 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.. SSIGNMNT GUI oncepts and Skills xplore xploring ngle-ngle Similarity in Triangles xample 1 Proving ngle-ngle Triangle Similarity xample pplying ngle-ngle Similarity Practice xercises 1 xercises 6 xercises or 1.5 _ or 1.5 _ or _ 1_ 1_ xample pplying SSS and SS for Triangle Similarity xercises etermine whether the two triangles are similar. If they are similar, write the similarity statement Houghton Mifflin Harcourt Publishing ompany y the Triangle ngle Sum Theorem, m 67. So and. y the Triangle Similarity Theorem,. 7 and are isosceles triangles, so and m 10, so m. However, m m 7. None of the angles in is congruent to, so the triangles are not similar. Module Lesson 4 xercise epth of Knowledge (.O.K.) Mathematical Practices 1 Skills/oncepts MP.5 Using Tools 17 Skills/oncepts MP. Reasoning 1 Skills/oncepts MP.4 Modeling 19 Strategic Thinking MP. Logic 617 Lesson 11.4

8 etermine whether the two triangles are similar. If they are similar, write the similarity statement , and by the Vertical ngles Theorem. Therefore by the Triangle Similarity Theorem. xplain how you know whether the triangles are similar. If possible, find the indicated length., and by the lternate Interior ngles Theorem. Therefore by the Triangle Similarity. VOI OMMON RRORS ecause they need to know only that two angles of two triangles are congruent in order to prove similarity, students might think they need to know only three angles of two quadrilaterals to do the same, and so on for any n-gon. Use counter examples to show that this is incorrect. Stress that triangles are a special case because they are rigid structures The triangles are similar by the Triangle Similarity Theorem. It is not possible to find the indicated length because, the length of the corresponding side,, is not known. The triangles are similar by Similarity. 1 5_ QR. ind. P Q 1. R The triangles are similar by Similarity. QR QR Not possible. Only one congruent angle is identified between and, so similarity cannot be established. Houghton Mifflin Harcourt Publishing ompany Module Lesson 4 xercise epth of Knowledge (.O.K.) Mathematical Practices 0 Strategic Thinking MP. Reasoning 1 Strategic Thinking MP. Logic Similarity of Triangles 61

9 INTGRT MTHMTIL PRTIS ocus on Patterns MP. When using the SSS and SS Similarity Theorems, some students have difficulty matching up the corresponding sides. Tell these students to match up the smallest side to the smallest side, the longest side to the longest side, and match the sides that are neither longest nor shortest. Show whether or not each pair of triangles are similar, if possible. Justify your answer, and write a similarity statement when the triangles are similar ; 5_ 9 9. The ratios are not equal, so the two triangles are not similar by the Vertical ngle _ Theorem _ 0.65; ( 4.0/6.4 ) Therefore by SS Similarity _.5;..5; Therefore by SSS Similarity. G 60 1 H J P Q R The triangles cannot be proven similar using the given information, because the congruent angle is not an included angle. Houghton Mifflin Harcourt Publishing ompany 15. xplain the rror student analyzes the two triangles shown below. xplain the error that the student makes , and _ ecause the two ratios are not equal, the two triangles are not similar. The student did not compare corresponding sides of the two triangles. is the shortest side of, so its corresponding side is _ the shortest side of. The ratios, _ and are equal, so the triangles are similar by SSS Similarity. Module Lesson Lesson 11.4

10 16. lgebra ind all possible values of x for which these two triangles are similar. x (x + ) (x - 50) 17. Multi-Step Identify two similar triangles in the figure, and explain why they are similar. Then find. 70 The possible values of x are the solutions of x 70, x x - 50, x + 70, x + x - 50, x + x , and x + x + + x These result in 50, 55, 60, or 70, of which only 50 results in similar triangles. So x 50 is the only possible value., and, so ~ by the Triangle Similarity Theorem 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 15 5'5" 65_ 4'6" h 1 inches 54 H.O.T. ocus on Higher Order Thinking 19. nalyze Relationships Prove the SS Triangle Similarity Theorem. Y Given: _ XY _ XZ and X Prove: XYZ Z X XY pply a dilation to with scale factor k and let the image of be '''. Then '. It is given that X, so by transitivity ' X. lso '' k XY XY and '' k XY XZ XZ. Therefore, '' XYZ by SS ongruence. So a sequence of rigid motions maps '' to XYZ. The dilation followed by this sequence of rigid motions shows that there is a sequence of similarity transformations that maps to XYZ. So XYZ. Houghton Mifflin Harcourt Publishing ompany Module Lesson 4 Similarity of Triangles 60

11 JOURNL 0. nalyze Relationships Prove the SSS Triangle Similarity Theorem. Have students write a journal entry to explain what a scale on a map means, how it is used, and how it is related to the concept of similarity. Given: _ XY _ XZ _ YZ Prove: XYZ (Hint: The main steps of the proof are similar to those of the proof of the Triangle X Similarity Theorem.) pply a dilation to with scale factor k XY and let the image of be '''. Then: '' k XY XY '' k XY XZ XZ '' k XY YZ YZ. Therefore, '' XYZ by the SSS ongruence Theorem. This means there is a sequence of similarity Y Z transformations that maps to XYZ. So XYZ. 1. 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. 6 X Houghton Mifflin Harcourt Publishing ompany Possible nswer: or X to be as small as possible, it should correspond to the shortest side of, which is. Thus, X corresponds to.. 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 the width and length of the model are each _ 1 of the corresponding dimensions of 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. triangle that lies entirely in a plane parallel to the ground will have side lengths that are 1 each _ of the side lengths of the corresponding triangle in the actual building. So the two 00 triangles are similar by the SSS Triangle Similarity Theorem. triangle that has vertices at several heights will not have side lengths that form a constant ratio with the side lengths of the corresponding triangle in the actual building, and so would not be similar. Module Lesson 4 61 Lesson 11.4

12 Lesson Performance Task ONNT VOULRY 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?. How do you know that is similar to G? G The name camera obscura comes from the Latin words camera, meaning room, and obscura, meaning dark. The plural of Latin words ending in the letter a is formed by changing a to ae. So, more than one of these instruments would be referred to as camerae obscurae.. Write a proportion you could use to find the height of the pine tree in., G in., 96 ft. How tall is the pine tree? 1. s the object gets closer the vertical angle at will get larger, and since remains similar to G, the height of the image on the back surface will increase. So the camera must be placed far enough from the object that the image is no taller than the height of the back surface, or else the object will not be able to be photographed.. Possible answer: G because the angles are alternate interior angles for parallel lines and G. G because vertical angles are congruent. So, G by the Similarity Theorem.. G ft 1 INTGRT MTHMTIL PRTIS ocus on Patterns MP. camera obscura is square. This means that in the figure in the Lesson Performance Task, G. Suppose you want to photograph an object that is n feet from the front of the camera. What is the maximum height of such an object if you want to photograph its entire height? xplain your reasoning. The maximum height is n feet. Sample answer: ecause G, and, _ 1. _ G, Houghton Mifflin Harcourt Publishing ompany Module 11 6 Lesson 4 XTNSION TIVITY Have students research methods of making simple models of a camera obscura. or some models, no more than a cereal box or shoe box, tape, scissors, and a pin are required. more elaborate model can be made by blocking a window of the classroom and projecting an outside scene onto a sheet in the classroom. ither method will give reasonable results when students compare the dimensions and angles of the external object with those of the projected image. Scoring Rubric points: Student correctly solves the problem and explains his/her reasoning. 1 point: 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Similarity of Triangles 6