Work with a partner. Use dynamic geometry software. a. Construct ABC and DEF with the side lengths given in column 1 of the table below.

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1 .3 roving riangle imilarity by and OMMO O Learning tandards HG-.. HG-..5 HG-G..5 HG-MG..1 OUIG VIL GUM o be proficient in math, you need to analyze situations by breaking them into cases and recognize and use counterexamples. ssential uestion What are two ways to use corresponding sides of two triangles to determine that the triangles are similar? eciding Whether riangles re imilar Work with a partner. Use dynamic geometry software. a. onstruct and with the side lengths given in column 1 of the table below m m m m m m b. opy the table and complete column 1. c. re the triangles similar? xplain your reasoning. d. epeat parts (a) (c) for columns 2 in the table. e. How are the corresponding side lengths related in each pair of triangles that are similar? Is this true for each pair of triangles that are not similar? f. Make a conjecture about the similarity of two triangles based on their corresponding side lengths. g. Use your conjecture to write another set of side lengths of two similar triangles. Use the side lengths to complete column 7 of the table. eciding Whether riangles re imilar Work with a partner. Use dynamic geometry software. onstruct any. a. ind,, and m. hoose any positive rational number k and construct so that = k, = k, and m = m. b. Is similar to? xplain your reasoning. c. epeat parts (a) and (b) several times by changing and k. escribe your results. ommunicate our nswer 3. What are two ways to use corresponding sides of two triangles to determine that the triangles are similar? ection.3 roving riangle imilarity by and 35

2 .3 Lesson What ou Will Learn ore Vocabulary revious similar figures corresponding parts slope parallel lines perpendicular lines Use the ide-ide-ide imilarity heorem. Use the ide-ngle-ide imilarity heorem. rove slope criteria using similar triangles. Using the ide-ide-ide imilarity heorem In addition to using congruent corresponding angles to show that two triangles are similar, you can use proportional corresponding side lengths. heorem heorem. ide-ide-ide () imilarity heorem If the corresponding side lengths of two triangles are proportional, then the triangles are similar. If = =, then. roof p. 37 IIG OI When using the imilarity heorem, compare the shortest sides, the longest sides, and then the remaining sides. Using the imilarity heorem Is either or GH similar to? OLUIO 1 9 H 10 ompare and by finding ratios of corresponding side lengths. hortest Longest emaining sides sides sides = = 1 1 = 9 G = 3 = 3 = 3 ll the ratios are equal, so. ompare and GH by finding ratios of corresponding side lengths. hortest Longest emaining sides sides sides GH = G = 1 1 H = 10 = 1 = 1 = 5 he ratios are not all equal, so and GH are not similar. 3 hapter imilarity

3 UIIG he arallel ostulate (ostulate 3.1) allows you to draw an auxiliary line in. here is only one line through point parallel to, so you are able to draw it. IIG OI ou can use either = or = in tep 1. Given K = KL = L rove KL imilarity heorem Locate on so that = K. raw so that. hen by the imilarity heorem (heorem.3), and = =. ou can use the given proportion and the fact that = K to deduce that = KL and = L. y the ongruence heorem (heorem 5.), it follows that KL. inally, use the definition of congruent triangles and the imilarity heorem (heorem.3) to conclude that KL. Using the imilarity heorem ind the value of x that makes. OLUIO x 1 1 3(x + 1) tep 1 ind the value of x that makes corresponding side lengths proportional. = = x 1 1 Write proportion. ubstitute. 1 = (x 1) ross roducts roperty 72 = x implify. 7 = x olve for x. tep 2 heck that the side lengths are proportional when x = 7. = x 1 = = 3(x + 1) = 2 =? = 1 =? K L = 2 When x = 7, the triangles are similar by the imilarity heorem. Monitoring rogress Use the diagram. 1. Which of the three triangles are similar? Write a similarity statement. 2. he shortest side of a triangle similar to is units long. ind the other side lengths of the triangle. Help in nglish and panish at igideasmath.com 2 X 20 M L ection.3 roving riangle imilarity by and 37

4 Using the ide-ngle-ide imilarity heorem heorem heorem.5 ide-ngle-ide () imilarity heorem If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. If X M and X M = X, then X M. M roof x. 33, p. 3 X M Using the imilarity heorem ou are building a lean-to shelter starting from a tree branch, as shown. an you construct the right end so it is similar to the left end using the angle measure and lengths shown? 53 ft 10 ft 53 9 ft 15 ft G H OLUIO oth m and m equal 53, so. ext, compare the ratios of the lengths of the sides that include and. horter sides G = 9 = 3 2 Longer sides H = = 3 2 he lengths of the sides that include and are proportional. o, by the imilarity heorem, GH. es, you can make the right end similar to the left end of the shelter. Monitoring rogress Help in nglish and panish at igideasmath.com xplain how to show that the indicated triangles are similar. 3.. XW X X W hapter imilarity

5 oncept ummary riangle imilarity heorems imilarity heorem imilarity heorem imilarity heorem If and, then. If = =, then. If and =, then. roving lope riteria Using imilar riangles ou can use similar triangles to prove the lopes of arallel Lines heorem (heorem 3.13). ecause the theorem is biconditional, you must prove both parts. 1. If two nonvertical lines are parallel, then they have the same slope. 2. If two nonvertical lines have the same slope, then they are parallel. he first part is proved below. he second part is proved in the exercises. art of lopes of arallel Lines heorem (heorem 3.13) Given n, and n are nonvertical. rove m = m n irst, consider the case where and n are horizontal. ecause all horizontal lines are parallel and have a slope of 0, the statement is true for horizontal lines. y n or the case of nonhorizontal, nonvertical lines, draw two such parallel lines, and n, and label their x-intercepts and, respectively. raw a vertical segment parallel to the y-axis from point on line to point on the x-axis. raw a vertical segment parallel to the y-axis from point on line n to point on the x-axis. ecause vertical and horizontal lines are perpendicular, and are right angles. x M O 1. n 1. Given orresponding ngles heorem (hm. 3.1) ight ngles ongruence heorem (hm. 2.3).. imilarity heorem (hm..3) 5. =. = 7. m =, m n = 5. orresponding sides of similar figures are proportional.. ewrite proportion. 7. efinition of slope. m n =. ubstitution roperty of quality 9. m = m n 9. ransitive roperty of quality ection.3 roving riangle imilarity by and 39

6 y n 0 hapter imilarity j k x o prove the lopes of erpendicular Lines heorem (heorem 3.1), you must prove both parts. 1. If two nonvertical lines are perpendicular, then the product of their slopes is If the product of the slopes of two nonvertical lines is 1, then the lines are perpendicular. he first part is proved below. he second part is proved in the exercises. art of lopes of erpendicular Lines heorem (heorem 3.1) Given n, and n are nonvertical. rove m m n = 1 raw two nonvertical, perpendicular lines, and n, that intersect at point. raw a horizontal line j parallel to the x-axis through point. raw a horizontal line k parallel to the x-axis through point on line n. ecause horizontal lines are parallel, j k. raw a vertical segment parallel to the y-axis from point to point on line k. raw a vertical segment parallel to the y-axis from point on line to point on line j. ecause horizontal and vertical lines are perpendicular, and are right angles. M O 1. n 1. Given 2. m = n 3. m = m + m 3. ngle ddition ostulate (ost. 1.). m + m = 90. ransitive roperty of quality lternate Interior ngles heorem (hm. 3.2). m = m. efinition of congruent angles 7. m + m = ubstitution roperty of quality. m = 90 m. olve statement 7 for m. 9. m + m + 90 = riangle um heorem (hm. 5.1) 10. m = 90 m 10. olve statement 9 for m. 11. m = m 11. ransitive roperty of quality.. efinition of congruent angles ight ngles ongruence heorem (hm. 2.3) imilarity heorem (hm..3) 15. = 1. = 17. m =, m n = 1. m m n = ( ) 19. m m n = ( 20. m m n = 1 ) 15. orresponding sides of similar figures are proportional. 1. ewrite proportion. 17. efinition of slope 1. ubstitution roperty of quality 19. ubstitution roperty of quality 20. implify.

7 .3 xercises ynamic olutions available at igideasmath.com Vocabulary and ore oncept heck 1. OML H ou plan to show that is similar to X by the imilarity heorem (heorem.). opy and complete the proportion that you will use: = =. 2. WHIH O O LOG? Which triangle does not belong with the other three? xplain your reasoning Monitoring rogress and Modeling with Mathematics In xercises 3 and, determine whether KL or is similar to. (ee xample 1.) 3.. In xercises 5 and, find the value of x that makes X. (ee xample 2.) x X 5x K 3(x 1) L K L X x In xercises 7 and, verify that. ind the scale factor of to. 7. : = 1, = 15, = : =, = 10, =. : = 10, = 1, = 20 : = 25, = 0, = 50 In xercises 9 and 10, determine whether the two triangles are similar. If they are similar, write a similarity statement and find the scale factor of triangle to triangle. (ee xample 3.) X 1 L K In xercises 11 and, sketch the triangles using the given description. hen determine whether the two triangles can be similar. 11. In, = 20, = 32, and m = 1. In GH, GH = 30, H =, and m H = W. he side lengths of are 2, x, and, and the side lengths of are 15, 25, and x. ection.3 roving riangle imilarity by and 1

8 In xercises 13 1, show that the triangles are similar and write a similarity statement. xplain your reasoning H 5 G K X V G 1. 1 In xercises 17 and 1, use X. 9 U IG O IIO In xercises 21 2, use the diagram to copy and complete the statement. L M m L = 22. m = 23. m = 2. = 25. m M = 2. m = 27. MKIG GUM our friend claims that KL MO by the imilarity heorem (heorem.5) when K = 1, m K = 130, KL = 1, M = 9, m = 5, and O =. o you support your friend s claim? xplain your reasoning. 2. LIG LIOHI ertain sections of stained glass are sold in triangular, beveled pieces. Which of the three beveled pieces, if any, are similar? 13 3 in. 3 in in. X 17. he shortest side of a triangle similar to X is 20 units long. ind the other side lengths of the triangle in. in. 7 in. in. 3 in. 3 in. 1. he longest side of a triangle similar to X is 39 units long. ind the other side lengths of the triangle. 19. O LI escribe and correct the error in writing a similarity statement by the imilarity heorem (heorem.5). 20. MHMIL OIO ind the value of n that makes X when =, = 5, X = (n + 1), = 7n 1, and. Include a sketch. 2 hapter imilarity 29. IG O IIO In the diagram, M M = M. Which of the statements must be true? M elect all that apply. xplain your reasoning. 2 1 M M M 30. WIIG re any two right triangles similar? xplain. 3

9 31. MOLIG WIH MHMI In the portion of the shuffleboard court shown, =. 35. WIIG an two triangles have all three ratios of corresponding angle measures equal to a value greater than 1? less than 1? xplain. 3. HOW O OU I? Which theorem could you use to show that O OM in the portion of the erris wheel shown when M = = 5 feet and MO = O = 10 feet? M a. What additional information do you need to show that using the imilarity heorem (heorem.)? O b. What additional information do you need to show that using the imilarity heorem (heorem.5)? 32. OO Given that is a right triangle and,, and are midpoints, prove that m = OVIG HOM Write a two-column proof of the imilarity heorem (heorem.5). Given, = rove 3. IIL HIKIG ou are given two right triangles with one pair of corresponding legs and the pair of hypotenuses having the same length ratios. a. he lengths of the given pair of corresponding legs are and 1, and the lengths of the hypotenuses are 10 and 30. Use the ythagorean heorem to find the lengths of the other pair of corresponding legs. raw a diagram. b. Write the ratio of the lengths of the second pair of corresponding legs. c. re these triangles similar? oes this suggest a Hypotenuse-Leg imilarity heorem for right triangles? xplain. 37. WIG OLUIO xplain why it is not necessary to have an ngle-ide-ngle imilarity heorem. 3. HOUGH OVOKIG ecide whether each is a valid method of showing that two quadrilaterals are similar. ustify your answer. a. b. c. d. 39. MULIL IO Use a diagram to show why there is no ide-ide-ngle imilarity heorem. 0. MOLIG WIH MHMI he dimensions of an actual swing set are shown. ou want to create a scale model of the swing set for a dollhouse using similar triangles. ketch a drawing of your swing set and label each side length. Write a similarity statement for each pair of similar triangles. tate the scale factor you used to create the scale model. ft 1 ft ft ft 1 in. 9 in. ft ft ection.3 roving riangle imilarity by and 3

10 1. OVIG HOM opy and complete the paragraph proof of the second part of the lopes of arallel Lines heorem (heorem 3.13) from page 39. Given m = m n, and n are nonvertical. y n rove n ou are given that m = m n. y the definition of slope, m = and m n =. y, =. ewriting this proportion yields. y the ight ngles ongruence heorem (hm. 2.3),. o, x by. ecause corresponding angles of similar triangles are congruent,. y, n. 2. OVIG HOM opy and complete the two-column proof of the second part of the lopes of erpendicular Lines heorem (heorem 3.1) from page 0. y n Given m m n = 1, and n are nonvertical. rove n j M O 1. m m n = 1 1. Given 2. m =, m n = 3.. = 5. = 2. efinition of slope = 1 3. Maintaining Mathematical roficiency. Multiply each side of statement 3 by. 5. ewrite proportion... ight ngles ongruence heorem (hm. 2.3) orresponding angles of similar figures are congruent lternate Interior ngles heorem (hm. 3.2) 10. m = m, m = m m + m + 90 = ubtraction roperty of quality 13. m + m = ubstitution roperty of quality 1. m = m + m 1. ngle ddition ostulate (ost. 1.) 15. m = efinition of perpendicular lines eviewing what you learned in previous grades and lessons ind the coordinates of point along the directed line segment so that to is the given ratio. (ection 3.5) 3. ( 3, ), (2, 1); 3 to 2. ( 3, 5), (9, 1); 1 to 3 5. (1, 2), (, ); to 3 k x hapter imilarity