8.3 Proving Triangle Similarity by SSS and SAS

EEssentialssential QQuestionuestion What are two ways to use corresponding sides of two triangles to determine that the triangles are similar?

Deciding Whether Triangles Are Similar 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.

1. 2. 3. 4. 5. 6. 7. AB 55615924 BC 888201218 AC 10 10 10 10 8 16 DE 10 15 9 12 12 8 EF 16 24 12 16 15 6 DF 20 30 15 8 10 8 m∠A m∠B m∠C m∠D m∠E m∠F CONSTRUCTING VIABLE ARGUMENTS b. Copy the table and complete column 1. To be profi cient in math, c. Are the triangles similar? Explain your reasoning. you need to analyze d. Repeat parts (a)–(c) for columns 2–6 in the table. situations by breaking them into cases and recognize e. How are the corresponding side lengths related in each pair of triangles that are and use counterexamples. 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.

Deciding Whether Triangles Are Similar Work with a partner. Use dynamic geometry software. Construct any △ABC. a. Find AB, AC, and m∠A. Choose any positive rational number k and construct △DEF so that DE = k ⋅ AB, DF = k ⋅ AC, and m∠D = m∠A. b. Is △DEF similar to △ABC? Explain your reasoning. c. Repeat parts (a) and (b) several times by changing △ABC and k. Describe your results. CCommunicateommunicate YourYour AnswerAnswer 3. What are two ways to use corresponding sides of two triangles to determine that the triangles are similar?

Section 8.3 Proving Triangle Similarity by SSS and SAS 435

hhs_geo_pe_0803.indds_geo_pe_0803.indd 435435 11/19/15/19/15 12:2412:24 PMPM 8.3 Lesson WWhathat YYouou WWillill LLearnearn Use the Side-Side-Side Similarity Theorem. Use the Side-Angle-Side Similarity Theorem. Core VocabularyVocabulary Prove slope criteria using similar triangles. Previous similar fi gures Using the Side-Side-Side Similarity Theorem corresponding parts slope In addition to using congruent corresponding angles to show that two triangles are similar, you can use proportional corresponding side lengths. parallel lines perpendicular lines TTheoremheorem Theorem 8.4 Side-Side-Side (SSS) Similarity Theorem If the corresponding side lengths of A R two triangles are proportional, then the triangles are similar. ST B C AB BC CA — = — = — △ABC ∼ △RST If RS ST TR, then .

Proof p. 437

Using the SSS Similarity Theorem

Is either △DEF or △GHJ similar to △ABC? FINDING AN B DF12 H ENTRY POINT 12 8 10 8 When using the 6 9 SSS Similarity Theorem, ACE JG 16 16 compare the shortest sides, the longest sides, and then SOLUTION the remaining sides. Compare △ABC and △DEF by fi nding ratios of corresponding side lengths. Shortest Longest Remaining sides sides sides AB 8 CA 16 BC 12 — = — — = — — = — DE 6 FD 12 EF 9 4 4 4 = — = — = — 3 3 3

All the ratios are equal, so △ABC ∼ △DEF.

Compare △ABC and △GHJ by fi nding ratios of corresponding side lengths. Shortest Longest Remaining sides sides sides AB 8 CA 16 BC 12 — = — — = — — = — GH 8 JG 16 HJ 10 6 = 1 = 1 = — 5

The ratios are not all equal, so △ABC and △GHJ are not similar.

436 Chapter 8 Similarity

hhs_geo_pe_0803.indds_geo_pe_0803.indd 436436 11/19/15/19/15 12:2412:24 PMPM SSS Similarity Theorem RS ST TR K S — = — = — Given JK KL LJ

Prove △RST ∼ △JKL J L P Q RT — — — — Locate P on RS so that PS = JK. Draw PQ so that PQ ʈ RT . Then △RST ∼ △PSQ by RS ST TR JUSTIFYING the AA Similarity Theorem (Theorem 8.3), and — = — = — . You can use the given STEPS PS SQ QP proportion and the fact that PS = JK to deduce that SQ = KL and QP = LJ. By the The Parallel Postulate SSS Congruence Theorem (Theorem 5.8), it follows that △PSQ ≅ △JKL. Finally, use (Postulate 3.1) allows you the defi nition of congruent triangles and the AA Similarity Theorem (Theorem 8.3) to to draw an auxiliary line conclude that △RST ∼ △JKL. PQ⃖៮៮⃗ in △RST. There is only one line through point P ⃖៮៮⃗ parallel to RT , so you are Using the SSS Similarity Theorem able to draw it. Find the value of x that makes △ABC ∼△DEF. E B 4 x − 1 12 18 A 8 C FINDING AN D 3(x + 1) F ENTRY POINT You can use either SOLUTION AB BC AB AC — = — — = — DE EF or DE DF Step 1 Find the value of x that makes corresponding side lengths proportional. in Step 1. AB BC — = — Write proportion. DE EF

4 x − 1 — = — Substitute. 12 18 4 ⋅ 18 = 12(x − 1) Cross Products Property 72 = 12x − 12 Simplify. 7 = x Solve for x. Step 2 Check that the side lengths are proportional when x = 7. BC = x − 1 = 6 DF = 3(x + 1) = 24 AB ? BC 4 6 AB ? AC 4 8 — = — — = — ✓ — = — — = — ✓ DE EF 12 18 DE DF 12 24

When x = 7, the triangles are similar by the SSS Similarity Theorem.

MMonitoringonitoring ProgressProgress Help in English and Spanish at BigIdeasMath.com R 24 S X Use the diagram. L 1. Which of the three triangles are similar? 30 33 39 36 Write a similarity statement. 20 26 T 2. The shortest side of a triangle similar M 24 N Y 30 Z to △RST is 12 units long. Find the other side lengths of the triangle.

Section 8.3 Proving Triangle Similarity by SSS and SAS 437

hhs_geo_pe_0803.indds_geo_pe_0803.indd 437437 11/19/15/19/15 12:2412:24 PMPM Using the Side-Angle-Side Similarity Theorem TTheoremheorem Theorem 8.5 Side-Angle-Side (SAS) Similarity Theorem If an angle of one triangle is congruent to an X M angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. P N ZX XY Z Y ∠X ≅ ∠M — = — △XYZ ∼ △MNP If and PM MN, then . Proof Ex. 33, p. 443

Using the SAS Similarity Theorem

YouY are building a lean-to shelter starting from a tree branch, as shown. Can you constructc the right end so it is similar to the left end using the angle measure and lengthsl shown?

F A 53° 10 ft 53° 15155ft ft 6 ft H C G 9 ft

B

SOLUTION Both m∠A and m∠F equal 53°, so ∠A ≅ ∠F. Next, compare the ratios of the lengths of the sides that include ∠A and ∠F. Shorter sides Longer sides AB 9 AC 15 — = — — = — FG 6 FH 10 3 3 = — = — 2 2 The lengths of the sides that include ∠A and ∠F are proportional. So, by the SAS Similarity Theorem, △ABC ∼ △FGH.

Yes, you can make the right end similar to the left end of the shelter.

MMonitoringonitoring ProgressProgress Help in English and Spanish at BigIdeasMath.com Explain how to show that the indicated triangles are similar.

3. △SRT ∼ △PNQ 4. △XZW ∼ △YZX S X P 20 15 24 18 12

NQ W 16 Z 9 Y R T 21 28

438 Chapter 8 Similarity

hhs_geo_pe_0803.indds_geo_pe_0803.indd 438438 11/19/15/19/15 12:2412:24 PMPM CConceptoncept SSummaryummary Triangle Similarity Theorems AA Similarity Theorem SSS Similarity Theorem SAS Similarity Theorem

A D A D A D

EF EF EF B C B C B C AB BC AC AB AC ∠A ≅ ∠D ∠B ≅ ∠E — = — = — ∠A ≅ ∠D — = — If and , If DE EF DF, then If and DE DF, then △ABC ∼ △DEF. △ABC ∼ △DEF. then △ABC ∼ △DEF.

Proving Slope Criteria Using Similar Triangles You can use similar triangles to prove the Slopes of Parallel Lines Theorem (Theorem 3.13). Because 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. The fi rst part is proved below. The second part is proved in the exercises.

Part of Slopes of Parallel Lines Theorem (Theorem 3.13)

Given ℓ ʈ n, ℓand n are nonvertical. m = m Prove ℓ n First, consider the case whereℓand n are horizontal. Because all horizontal lines are parallel and have a slope of 0, the statement is true for horizontal lines. y For the case of nonhorizontal, nonvertical lines, draw two such parallel lines,ℓand n, n x A D BC— E and label their -intercepts and , respectively. Draw a vertical segment parallel to— the y-axis from point B on lineℓto point C on the x-axis. Draw a vertical segment B EF parallel to the y-axis from point E on line n to point F on the x-axis. Because vertical and horizontal lines are perpendicular, ∠BCA and ∠EFD are right angles.

A D STATEMENTS REASONS C F x 1. ℓ ʈ n 1. Given 2. ∠BAC ≅ ∠EDF 2. Corresponding Angles Theorem (Thm. 3.1) 3. ∠BCA ≅ ∠EFD 3. Right Angles Congruence Theorem (Thm. 2.3) 4. △ABC ∼ △DEF 4. AA Similarity Theorem (Thm. 8.3) BC AC — = — 5. EF DF 5. Corresponding sides of similar fi gures are proportional. BC EF — = — 6. AC DF 6. Rewrite proportion. BC EF m = — m = — 7. ℓ AC , n DF 7. Defi nition of slope BC m = — 8. n AC 8. Substitution Property of Equality m = m 9. ℓ n 9. Transitive Property of Equality

Section 8.3 Proving Triangle Similarity by SSS and SAS 439

hhs_geo_pe_0803.indds_geo_pe_0803.indd 439439 11/19/15/19/15 12:2412:24 PMPM To prove the Slopes of Perpendicular Lines Theorem (Theorem 3.14), you must prove both parts. 1. If two nonvertical lines are perpendicular, then the product of their slopes is −1. 2. If the product of the slopes of two nonvertical lines is −1, then the lines are perpendicular. The fi rst part is proved below. The second part is proved in the exercises.

Part of Slopes of Perpendicular Lines Theorem (Theorem 3.14) Given ℓ⊥ n,ℓand n are nonvertical. mm = − Prove ℓ n 1 y n E Draw two nonvertical, perpendicular lines,ℓand n, that intersect at point A. Draw a A horizontal line j parallel to the x-axis through point A. Draw a horizontal line k parallel to the x-axis through point C on line n. Because horizontal lines are parallel, j ʈ k. D j AB— y A B k Draw a vertical segment — parallel to the -axis from point to point on line . Draw a vertical segment ED parallel to the y-axis from point E on lineℓto point D on BC k line j. Because horizontal and vertical lines are perpendicular, ∠ ABC and ∠ ADE are x right angles. STATEMENTS REASONS 1. ℓ⊥ n 1. Given 2. m∠CAE = 90° 2. ℓ⊥ n 3. m∠CAE = m∠DAE + m∠CAD 3. Angle Addition Postulate (Post. 1.4) 4. m∠DAE + m∠CAD = 90° 4. Transitive Property of Equality 5. ∠BCA ≅ ∠CAD 5. Alternate Interior Angles Theorem (Thm. 3.2) 6. m∠BCA = m∠CAD 6. Defi nition of congruent angles 7. m∠DAE + m∠BCA = 90° 7. Substitution Property of Equality 8. m∠DAE = 90° − m∠BCA 8. Solve statement 7 for m∠DAE. 9. m∠BCA + m∠BAC + 90° = 180° 9. Triangle Sum Theorem (Thm. 5.1) 10. m∠BAC = 90° − m∠BCA 10. Solve statement 9 for m∠BAC. 11. m∠DAE = m∠BAC 11. Transitive Property of Equality 12. ∠DAE ≅ ∠BAC 12. Defi nition of congruent angles 13. ∠ABC ≅ ∠ADE 13. Right Angles Congruence Theorem (Thm. 2.3) 14. △ABC ∼ △ADE 14. AA Similarity Theorem (Thm. 8.3) AD DE — = — 15. AB BC 15. Corresponding sides of similar fi gures are proportional. AD AB — = — 16. DE BC 16. Rewrite proportion. DE AB m = — m = −— 17. ℓ AD, n BC 17. Defi nition of slope DE AB m m = — −— 18. ℓ n AD ⋅ ( BC ) 18. Substitution Property of Equality DE AD m m = — −— 19. ℓ n AD ⋅ ( DE ) 19. Substitution Property of Equality m m = − 20. ℓ n 1 20. Simplify. 440 Chapter 8 Similarity

hhs_geo_pe_0803.indds_geo_pe_0803.indd 440440 11/19/15/19/15 12:2412:24 PMPM 8.3 Exercises Dynamic Solutions available at BigIdeasMath.com

VVocabularyocabulary andand CoreCore ConceptConcept CheckCheck 1. COMPLETE THE SENTENCE You plan to show that △QRS is similar to △XYZ by the SSS Similarity Theorem QR QS — = — = — (Theorem 8.4). Copy and complete the proportion that you will use: YZ .

2. WHICH ONE DOESN’T BELONG? Which triangle does not belong with the other three? Explain your reasoning.

8 9 12 6 8 6 4 4 3 6 12 18

MMonitoringonitoring ProgressProgress andand ModelingModeling withwith MathematicsMathematics

In Exercises 3 and 4, determine whether △JKL or In Exercises 7 and 8, verify that △ABC ∼ △DEF. Find △RST is similar to △ABC. (See Example 1.) the scale factor of △ABC to △DEF.

3. B 8 C K 74L S T 7. △ABC: BC = 18, AB = 15, AC = 12 △DEF EF = DE = DF = 3.5 : 12, 10, 8 6 6 7 11 12 R 8. △ABC: AB = 10, BC = 16, CA = 20 J △DEF: DE = 25, EF = 40, FD = 50 A In Exercises 9 and 10, determine whether the two 4. L triangles are similar. If they are similar, write a B T similarity statement and fi nd the scale factor of 25 triangle B to triangle A. (See Example 3.) 16 14 17.5 16 10.5 Y 9. DF12 AJ20 C KR20 12 S A 8 9 In Exercises 5 and 6, fi nd the value of x that makes B △DEF ∼ △XYZ. (See Example 2.) E X 6 W 5. Y E 10. S 2x − 1 10 14 5 A 112° 8 DFX11 5x + 2 Z L 24 J R 18 T 112° 10 6. E 8 F B K X 10 In Exercises 11 and 12, sketch the triangles using the x − 7.5 3( 1) x − 1 given description. Then determine whether the two triangles can be similar. ZY4 D 11. In △RST, RS = 20, ST = 32, and m∠S = 16°. In △FGH, GH = 30, HF = 48, and m∠H = 24°.

12. The side lengths of △ABC are 24, 8x, and 48, and the side lengths of △DEF are 15, 25, and 6x.

Section 8.3 Proving Triangle Similarity by SSS and SAS 441

hhs_geo_pe_0803.indds_geo_pe_0803.indd 441441 11/19/15/19/15 12:2412:24 PMPM In Exercises 13–16, show that the triangles are ATTENDING TO PRECISION In Exercises 21–26, use the similar and write a similarity statement. Explain diagram to copy and complete the statement. your reasoning. L 3.5 S 2 13. F 14. E M 2 5 14 4 44.5° G 27 24 A D N C 18 15 61° 18 21 4 8 H 16.5J 5.5 K B P 91° 4 R 15. XZ30 J Q 47° 21 35 21. m∠LNS = 22. m∠NRQ = Y 47° GD50 23. m∠NQR = 24. RQ = 16. R 12 S m∠NSM = m∠NPR = T 25. 26.

16 18 27. MAKING AN ARGUMENT Your friend claims that 24 12 △JKL ∼ △MNO by the SAS Similarity Theorem (Theorem 8.5) when JK = 18, m∠K = 130°, V U Q 9 KL = 16, MN = 9, m∠N = 65°, and NO = 8. Do you support your friend’s claim? Explain your reasoning. In Exercises 17 and 18, use △XYZ. 28. ANALYZING RELATIONSHIPS Certain sections of Y stained glass are sold in triangular, beveled pieces. Which of the three beveled pieces, if any, are similar?

13 12 3 in. 3 in. 5.25 in. 3 in. 5 in. X 10 Z 3 in. 7 in. 17. The shortest side of a triangle similar to △XYZ is 20 units long. Find the other side lengths of the 4 in. triangle. 4 in.

18. The longest side of a triangle similar to △XYZ is 29. ATTENDING TO PRECISION In the diagram, 39 units long. Find the other side lengths of the MN MP — = — triangle. MR MQ. Which of the statements must be true? Select all that apply. Explain your reasoning. 19. ERROR ANALYSIS Describe and correct the error in writing a similarity statement. Q R 2 3 B PR A 15 N 14P ✗ 86° 18 M 2486° 20 Q — — C ○A ∠1 ≅ ∠2 ○B QR ʈ NP △ABC ∼ △PQR by the SAS C ∠ ≅ ∠ D △MNP ∼ △MRQ Similarity Theorem (Theorem 8.5). ○ 1 4 ○

30. WRITING Are any two right triangles similar? 20. MATHEMATICAL CONNECTIONS Find the value of n Explain. that makes △DEF ∼ △XYZ when DE = 4, EF = 5, XY = 4(n + 1), YZ = 7n − 1, and ∠E ≅ ∠Y. Include a sketch. 442 Chapter 8 Similarity

hhs_geo_pe_0803.indds_geo_pe_0803.indd 442442 11/19/15/19/15 12:2412:24 PMPM 31. MODELING WITH MATHEMATICS In the portion of 35. WRITING Can two triangles have all three ratios of BC BD corresponding angle measures equal to a value greater the shuffl eboard court shown, — = — . AC AE than 1? less than 1? Explain.

C 36. HOW DO YOU SEE IT? Which theorem could you D use to show that △OPQ ∼ △OMN in the portion of PM = QN = B the Ferris wheel shown when 5 feet E and MO = NO = 10 feet?

A P M a. What additional information do you need to show △BCD ∼ △ACE Q that using the SSS Similarity O Theorem (Theorem 8.4)? N b. What additional information do you need to show that △BCD ∼ △ACE using the SAS Similarity Theorem (Theorem 8.5)?

32. PROOF Given that △BAC is a right triangle and D, E, 37. DRAWING CONCLUSIONS Explain why it is not and F are midpoints, prove that m∠DEF = 90°. necessary to have an Angle-Side-Angle Similarity B Theorem. E D 38. THOUGHT PROVOKING Decide whether each is a A F C valid method of showing that two quadrilaterals are similar. Justify your answer.

33. PROVING A THEOREM Write a two-column proof of a. SASA b. SASAS c. SSSS d. SASSS the SAS Similarity Theorem (Theorem 8.5). AB AC 39. MULTIPLE REPRESENTATIONS Use a diagram to show Given ∠A ≅ ∠D, — = — DE DF why there is no Side-Side-Angle Similarity Theorem. Prove △ABC ∼ △DEF 40. MODELING WITH MATHEMATICS The dimensions B E of an actual swing set are shown. You want to create a scale model of the swing set for a dollhouse using similar triangles. Sketch a drawing of your swing D F set and label each side length. Write a similarity A C statement for each pair of similar triangles. State the scale factor you used to create the scale model. 34. CRITICAL THINKING You are given two right triangles with one pair of corresponding legs and the pair of 14 ft hypotenuses having the same length ratios. D A a. The lengths of the given pair of corresponding legs are 6 and 18, and the lengths of the hypotenuses 6 ft are 10 and 30. Use the Pythagorean Theorem to 6 ft fi nd the lengths of the other pair of corresponding 8 ft 8 ft legs. Draw a diagram. F E b. Write the ratio of the lengths of the second pair 18 in. of corresponding legs. 9 in. B c. Are these triangles similar? Does this suggest a C Hypotenuse-Leg Similarity Theorem for right 4 ft triangles? Explain.

Section 8.3 Proving Triangle Similarity by SSS and SAS 443

hhs_geo_pe_0803.indds_geo_pe_0803.indd 443443 11/19/15/19/15 12:2412:24 PMPM 41. PROVING A THEOREM Copy and complete the paragraph proof of the second part of y B n the Slopes of Parallel Lines Theorem (Theorem 3.13) from page 439. E m = m n Given ℓ n,ℓand are nonvertical. Prove ℓ ʈ n BC EF m = m . m = — m = — You are given that ℓ n By the defi nition of slope, ℓ AC and n DF. By A D BC EF C F x — = — ______, AC DF. Rewriting this proportion yields ______. By the Right Angles Congruence Theorem (Thm. 2.3), ______. So, △ABC ∼ △DEF by ______. Because corresponding angles of similar triangles are congruent, ∠BAC ≅ ∠EDF. By ______, ℓ ʈ n.

42. PROVING A THEOREM Copy and complete the two-column proof of the second part of y n the Slopes of Perpendicular Lines Theorem (Theorem 3.14) from page 440. E m m = − n Given ℓ n 1, ℓand are nonvertical. A Prove ℓ⊥ n D j

STATEMENTS REASONS BC k m m = − x 1. ℓ n 1 1. Given DE AB m = — m = −— 2. ℓ AD, n BC 2. Defi nition of slope DE AB — −— = − 3. AD ⋅ BC 1 3. ______DE BC — = — 4. AD AB 4. Multiply each side of statement 3 BC −— by AB. DE — = — 5. BC 5. Rewrite proportion. 6. ______6. Right Angles Congruence Theorem (Thm. 2.3) 7. △ABC ∼ △ADE 7. ______8. ∠BAC ≅ ∠DAE 8. Corresponding angles of similar fi gures are congruent. 9. ∠BCA ≅ ∠CAD 9. Alternate Interior Angles Theorem (Thm. 3.2) 10. m∠BAC = m∠DAE, m∠BCA = m∠CAD 10. ______11. m∠BAC + m∠BCA + 90° = 180° 11. ______12. ______12. Subtraction Property of Equality 13. m∠CAD + m∠DAE = 90° 13. Substitution Property of Equality 14. m∠CAE = m∠DAE + m∠CAD 14. Angle Addition Postulate (Post. 1.4) 15. m∠CAE = 90° 15. ______16. ______16. Defi nition of perpendicular lines

MMaintainingaintaining MathematicalMathematical ProficiencyProficiency Reviewing what you learned in previous grades and lessons Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. (Section 3.5)

43. A(−3, 6), B(2, 1); 3 to 2 44. A(−3, −5), B(9, −1); 1 to 3 45. A(1, −2), B(8, 12); 4 to 3

444 Chapter 8 Similarity

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