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7-3 7-3 Exercises Exercises KEYWORD: MG7 7-3 KEYWORD: MG7 Parent GUIDED PRACTICE Assignment Guide SEE EXAMPLE Assign Guided Practice exercises as necessary. 1 p. 470 Explain why the triangles are similar and write a similarity statement. 1. * 2. If you finished Examples 1–3 Basic 11–16, 20–24, 31 Average 11–16, 19–24, 27, 31 Advanced 11–16, 20–24, 27, 31, 37, 40 xÓ SEE EXAMPLE 2 n£Â xÓ If you finished Examples 1–5 Basic 11–19, 23–25, 32–37, 41–46 Average 11–25, 29–37, 41–46 Advanced 11–20, 23–28, 30, 31, 33–46 n 4. MNP and MRQ Óä { È , Multi-Step Explain why the triangles are similar and then find each length. 5. AB 6. WY Quickly check key concepts. Exercises: 11, 12, 14, 16, 18, 23 1 SEE EXAMPLE 4 1. By the Sum Thm., m∠ A = 47°. So by the def. of , ∠ A ∠F, and ∠C ∠H. Therefore ABC ∼ FGH by AA ∼. £x 7 6 Ç 9 −− 7. Given: MN KL 8. Given: SQ = 2QP, TR = 2RP Prove: JMN ∼ JKL p. 472 Prove: PQR ∼ PST * 2. It is given that ∠P ∠T. ∠QST is a rt. ∠ by the Lin. Pair Thm., so ∠QST ∠RSP. Therefore QST ∼ RSP by AA ∼. DE EF 1 3. DF = = = , so DEF 2 JK KL JL ∼ JKL by SSS ∼. + , - / 9. The coordinates of A, B, and C are A(0, 0), B(2, 6), and C(8, -2). What theorem or postulate justifies the statement ABC ∼ ADE, if the coordinates of D and E are twice the coordinates of B and C? SAS or SSS ∼ Thm. _ _ _ _ 4. It is given that ∠NMP 2 MN MP ∠RMQ. ___ = ___ = __ . Therefore 3 MR MQ MNP ∼ MRQ by SAS ∼. 8 n°Çx È Answers + * n £Ó 3 { £È £ä È p. 471 , - Verify that the triangles are similar. SEE EXAMPLE / 3. DEF and JKL p. 471 Homework Quick Check + {Ç SEE EXAMPLE p. 472 5. It is given that ∠ AED ∠ ACB. ∠ A ∠ A by the Reflex. Prop. of . Therefore AED ∼ ACB by AA ∼. AB = 10 5 10. Surveying In order to measure the distance AB across the meteorite crater, a surveyor at S locates points A, B, C, and D as shown. What is AB to the nearest meter? nearest kilometer? A 733 m C 586 m 533 m S 1200 m, or 1.2 km 644 m D 6–8. See p. A23. 800 m 474 B Chapter 7 Similarity *À>VÌViÊ 4RIANGLE3IMILARITY!!3333!3 7-3 PRACTICE A ,%33/. ÇÎ &ILLINTHEBLANKSTOCOMPLETEEACHPOSTULATEORTHEOREM PROPORTIONAL )FTHETHREESIDESOFONETRIANGLEARE ANOTHERTRIANGLETHENTHETRIANGLESARESIMILAR TOTHETHREESIDESOF )FTWOSIDESOFONETRIANGLEAREPROPORTIONALTOTWOSIDESOFANOTHERTRIANGLEANDTHEIR ge07se_c07_0470_0477.indd 474 SIMILAR INCLUDEDANGLESARECONGRUENTTHENTHETRIANGLESARE )FTWOANGLESOFONETRIANGLEARE TRIANGLETHENTHETRIANGLESARESIMILAR CONGRUENT 5/11/06 4:16:33 PM TOTWOANGLESOFANOTHER .AMETWOPAIRSOFCONGRUENTANGLESIN%XERCISESANDTOSHOWTHAT THETRIANGLESARESIMILARBYTHE!NGLE!NGLE!!3IMILARITY0OSTULATE ! 8 37° 4 $ 37° : " 9 # 5 "9 #: 6 & % &6 $4 3UBSTITUTESIDELENGTHSINTOTHERATIOSIN%XERCISE)FTHERATIOSAREEQUAL THETRIANGLESARESIMILARBYTHE3IDE3IDE3IDE3333IMILARITY4HEOREM 12 ' 6 12 ) + ( 9 * 24 18 , ??? '( ??? *+ ?? ??? () +, ??? ?? ??? ') *, ??? ?? .AMEONEPAIROFCONGRUENTANGLESANDSUBSTITUTESIDELENGTHSINTO THERATIOSIN%XERCISE)FTHERATIOSAREEQUALANDTHECONGRUENT ANGLESAREINBETWEENTHEPROPORTIONALSIDESTHETRIANGLESARESIMILAR BYTHE3IDE!NGLE3IDE3!33IMILARITY4HEOREM KEYWORD: MG7 Resources 0 474 Chapter 7 5 8 1 20 4 10 16 2 CONGRUENTANGLES 3 14 ??? 01 34 ??? ?? ??? 12 45 ??? ?? PRACTICE AND PROBLEM SOLVING 11–12 13–14 15–16 17–18 19 "" Explain why the triangles are similar and write a similarity statement. Independent Practice For See Exercises Example 11. 12. 1 2 3 4 5 A common error in Exercise 16 is to let PS = x and then to write and 10 x solve the proportion __ = ____ . Point 6 17.5 out that 6 is not the length of a side of any triangle, so it cannot be used in the proportion. Encourage students to draw PST and PVW separately and then write the correct 10 x proportion, _____ = ____ . x+6 17.5 Ç{ ÎÓ Verify that the given triangles are similar. Extra Practice 13. KLM and KNL Skills Practice p. S16 Application Practice p. S34 14. UVW and XYZ , £ ÊÊÊÊ xÊÊÚÚ 8 1 Ó £ ÊÊÊÊ xÊÊÚÚ { { 9 Ó 11. It is given . + £ 7 6 that ∠GLH ∠K. xÊÊÚÚÊÊÊÊ { Ó < ∠G ∠G by the Reflex. Prop. of . Multi-Step Explain why the triangles are similar and then find each length. Therefore HLG ∼ * 15. AB 16. PS JKG by AA ∼. 12. By the Isosc. Thm., ∠C ∠B. { £Ó By the Sum Thm. m∠C = m∠B = 74°. In the same way, 17. Given: CD = 3AC, CE = 3BC m∠F = 74°. So Prove: ABC ∼ DEC by the def. of , ∠B ∠E and ∠C ∠F. Therefore ABC ∼ DEF by AA ∼. È ZX Algebra In Exercise 16, remind students that in any proportion such 10 x as _____ = ____ , they should use x+6 17.5 parentheses to write the product as 10(x + 6). Then use the Distributive Property to remove the parentheses. / £ä 7 £Ç°x Answers 15. It is given that ∠ ABD ∠C. ∠ A ∠ A by the Reflex. Prop. of . Therefore ABD ∼ ACB by AA ∼. AB = 8 −− −− 16. Since ST VW, ∠PST ∠V by the Corr. Post. ∠P ∠P by the Reflex. Prop. of . Therefore PST ∼ PVW by AA ∼. PS = 8 QR PR = _ 18. Given: _ MR NR Prove: ∠1 ∠2 £ * + Ó , 13. ∠K ∠K by the Reflex. Prop. of 19. Photography The picture shows a KL KM person taking a pinhole photograph . ___ = ___ = __3 . KN KL 2 of himself. Light entering the opening Therefore KLM ∼ reflects his image on the wall, forming KNL by SAS ∼. similar triangles. What is the height UV VW 14. ___ = ___ XY YZ WU 8 = ___ = __ . - 6 17. 1. CD = 3AC, CE = 3BC (Given) CD CE 2. = 3, = 3 (Div. Prop. AC BC of =) 3. ∠ ACB ∠DCE (Vert. Thm.) 4. ABC ∼ DEC (SAS ∼ Steps 2, 3) _ 15 in. 5 ft 5 in. 4 ft 6 in. of the image to the nearest tenth of a foot? 1.5 ft PR = 18. 1. ___ MR 11 Therefore UVW ∼ XYZ by SSS ∼. Draw JKL and MNP. Determine if you can conclude that JKL ∼ MNP based on the given information. If so, which postulate or theorem justifies your response? JK JL JL JK KL KL = _ KL 21. _ = _ 22. ∠J ∠M, _ = _ 20. ∠K ∠N, _ = _ MN NP MN NP MP MP NP no yes; SAS ∼ yes; SSS ∼ Find the value of x. 23. Î + Ý * 24. 3 - _ QR ___ (Given) NR 2. ∠R ∠R (Reflex. Prop. of ) 3. PQR ∼ MNR (SAS ∼ Steps 1, 2) 4. ∠1 ∠2 (Def. of ∼ ) 7 ÝÊÊ Óä ÝÊÊx , ÓÝÊÊÓ { Ê,,", ,/ £x / 7- 3 Triangle Similarity: AA, SSS, and SAS ,%33/. ÇÎ *À>VÌViÊ 4RIANGLE3IMILARITY!!3333!3 7-3 PRACTICE B ,%33/. ÇÎ $ ! # 4 7 8 5 ge07se_c07_0470_0477.indd 475 0OSSIBLEANSWER!#"AND 6 9 0OSSIBLEANSWER%VERYEQUILATERAL %#$ARECONGRUENTVERTICAL TRIANGLEISALSOEQUIANGULARSO ANGLESM"M$ EACHANGLEINBOTHTRIANGLES SO"$4HUSN!"# MEASURES 4HUSN456 N%$#BY!! N789BY!! N*,+ANDN*-. N012ANDN543 + 2.1 0 16 &IND34 1.8 2 3 8 12 0OSSIBLEANSWER)TISGIVENTHAT N*-.,??? +,??? *,?? -. *- 4HUSN*+,N*-.BY3!3 3 4 3.5 0 4 4HUSN012N543BY333 % 5 3.25 # $ 1.25 ' & 0OSSIBLEANSWER##BYTHE2EFLEXIVE0ROPERTY#'$AND&ARE RIGHTANGLESSOTHEYARECONGRUENT4HUSN#$'N#%&BY!! $% # $ ^ 5/19/07 1:48:44 PM ^ q q 3 5 7 2 5 6 0OSSIBLEANSWER??? 01??? 12?? 54 43 &OR%XERCISEEXPLAINWHYTHETRIANGLESARE SIMILARANDFINDTHESTATEDLENGTH $% ??? 2 4 3 1 6 5SETRIANGLESIMILARITYTOPROVETHAT'()*+01234 ? ? ? 0OSSIBLEANSWER$RAWDIAGONALS(+(*13 . 1 3.6 , ! ALSOEACHHAVEARIGHTANGLESON!"#N!$"BY!!4HEYHAVEA N!"#ANDN"$#SHARE#4HEYALSOEACHHAVE SIMILARITYRATIOOF?? ARIGHTANGLESON!"#N"$#BY!!4HEYHAVEASIMILARITYRATIO ^ q OF???? "YTHE4RANSITIVE0ROPERTYOF3IMILARITYN!$"N"$#4HEY ^ q HAVEASIMILARITYRATIOOF??? ^ q !$AND$#&INDTHEPERIMETEROFN!"# 5SETHESIMILARITYRATIOSYOUFOUNDIN%XERCISEANDTHEANSWER TO%XERCISETOFINDTHEPERIMETERSOFN!$"ANDN"$# &OR%XERCISESANDVERIFYTHATTHETRIANGLESARESIMILAR%XPLAINWHY 24 " 0OSSIBLEANSWERN!"#ANDN!$"SHARE!4HEY % " - 7-3 PRACTICE C 0ROVESIMILARITYRELATIONSHIPSBETWEENTRIANGLESINTHEFIGURE 'IVEASIMILARITYRATIOFOREACHRELATIONSHIPYOUFIND * *À>VÌViÊ 4RIANGLE3IMILARITY!!3333!3 5SETHEFIGUREFOR%XERCISESn &OR%XERCISESANDEXPLAINWHYTHETRIANGLESARESIMILARANDWRITE ASIMILARITYSTATEMENT 475 ? AND14'AND0ARERIGHTANGLESSOTHEYARE ) ( * ' + 4 3 0 2 1 SON'(+N014BY3!3)TISGIVENTHAT CONGRUENT??? '+??? '(?? 04 01 SON()*N123BY3!3"ECAUSEN'(+ )2??? ()??? )*?? 12 23 N014??? (+?? AND'(+014"ECAUSEN()*N123??? (*?? 14 13 AND)(*213)TISGIVENTHAT(13OBYTHE!NGLE!DDITION SON+(*N413BY3!3 0OSTULATE+(*413??? (+??? (*?? 14 13 "ECAUSEN+(*N413??? +*??? (+?? !LLTHECORRESPONDINGANGLESARE 34 14 CONGRUENTALLTHECORRESPONDINGSIDESAREPROPORTIONAL4HUS'()*+ 01234 Lesson 7-3 475