UNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 7: Proving Similarity Instruction

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1 UNIT 1 SIMILRITY, ONGRUENE, N PROOFS Lsson 7: Proving Similrity Prrquisit Skills This lsson rquirs th us of th following skills: using th distn formul to find th lngths of sids of tringls ing fmilir with th Pythgorn Thorm working with nd simplifying squr roots idntifying similr tringls using similrity sttmnts to find unknown lngths nd msurs of similr tringls Introdution Gomtry inluds mny dfinitions nd sttmnts. On sttmnt hs n shown to tru, it is lld thorm. Thorms, lik dfinitions, n usd to show othr sttmnts r tru. On of th most wll known thorms of gomtry is th Pythgorn Thorm, whih rlts th lngth of th hypotnus of right tringl to th lngths of its lgs. Th thorm stts tht th sum of th squrs of th lngths of th lgs ( nd ) of right tringl is qul to th squr of th lngth of th hypotnus (). This n writtn lgrilly s Th Pythgorn Thorm hs mny pplitions nd n vry hlpful whn solving rl-world prolms. Thr r svrl wys to prov th Pythgorn Thorm; on wy is y using similr tringls nd similrity sttmnts. Ky onpts Th Pythgorn Thorm Th Pythgorn Thorm is oftn usd to find th lngths of th sids of right tringl, tringl tht inluds on 90 ngl. Thorm Pythgorn Thorm Th sum of th squrs of th lngths of th lgs ( nd ) of right tringl is qul to th squr of th lngth of th hypotnus (). Wlh Edution U1-415 GPS nlyti Gomtry Thr Rsour

2 UNIT 1 SIMILRITY, ONGRUENE, N PROOFS Lsson 7: Proving Similrity In th tringl on th prvious pg, ngl is 90, s shown y th squr. Th longst sid of th right tringl,, is lld th hypotnus nd is lwys lotd ross from th right ngl. Th lgs of th right tringl, nd, r th two shortr sids. It is lso tru tht if th sum of th squrs of th msurs of two sids of tringl quls th squr of th msur of th longst sid, thn th tringl is right tringl. This is known s th onvrs of th Pythgorn Thorm. To prov th Pythgorn Thorm using similr tringls, you must first idntify th similr tringls. In this xmpl, thr is only on tringl givn. gin y drwing th ltitud, th sgmnt from ngl tht is prpndiulr to th lin ontining th opposit sid,. Noti tht y rting th ltitud, w hv rtd two smllr right tringls, nd, within th lrgr givn right tringl,. nd r 90 nd r thrfor ongrunt. of is ongrunt to of us of th Rflxiv Proprty of ongrun. U1-416 GPS nlyti Gomtry Thr Rsour Wlh Edution

3 UNIT 1 SIMILRITY, ONGRUENE, N PROOFS Lsson 7: Proving Similrity ording to th ngl-ngl () Similrity Sttmnt, if two ngls of on tringl r ongrunt to two ngls of nothr tringl, thn th tringls r similr; thrfor,. nd r 90 nd r thrfor ongrunt. of is ongrunt to of us of th Rflxiv Proprty of ongrun. Two ngls in r ongrunt to two ngls in ; thrfor,. Similrity is trnsitiv. Sin nd, thn. d orrsponding sids of similr tringls r proportionl; thrfor, d nd. trmining th ross produts of h proportion lds to th Pythgorn Thorm. d d 2 2 d ( + d) dd oth qutions. Ftor. ( + d) is qul to us of sgmnt ddition. Wlh Edution U1-417 GPS nlyti Gomtry Thr Rsour

4 UNIT 1 SIMILRITY, ONGRUENE, N PROOFS Lsson 7: Proving Similrity Th onvrs of th Pythgorn Thorm n usful whn proving right tringls using similr tringls. Typs of Proofs Prgrph proofs r sttmnts writtn out in omplt sntns in logil ordr to show n rgumnt. Flow proofs r grphil mthod of prsnting th logil stps usd to show n rgumnt. In flow proof, th logil sttmnts r writtn in oxs nd th rson for h sttmnt is writtn low th ox. nothr ptd form of proof is two-olumn proof. Two-olumn proofs inlud numrd sttmnts nd orrsponding rsons tht show th rgumnt in logil ordr. Two-olumn proofs ppr in th Guidd Prti xmpls tht follow. ommon Errors/Misonptions misidntifying th ltituds of tringls inorrtly simplifying xprssions with squr roots U1-418 GPS nlyti Gomtry Thr Rsour Wlh Edution

5 UNIT 1 SIMILRITY, ONGRUENE, N PROOFS Lsson 7: Proving Similrity Guidd Prti Exmpl 1 Writ two-olumn proof to prov th Pythgorn Thorm using similr tringls. 1. rw in hlpful informtion. rw th ltitud from nd ll th point of intrstion on s. Ll s. Ll s f. f Wlh Edution U1-419 GPS nlyti Gomtry Thr Rsour

6 UNIT 1 SIMILRITY, ONGRUENE, N PROOFS Lsson 7: Proving Similrity 2. Idntify th similr tringls. It is oftn hlpful to rdrw th tringls. f 3. rt th two-olumn proof. Sttmnts 1. with right f ; 4. f 2 ; 2 5. f ( f + ) f 8. () or Rsons 1. Givn 2. If th ltitud is drwn to th hypotnus of right tringl, thn th two tringls formd r similr to th originl tringl nd h othr. 3. finition of similr tringls; orrsponding sids r proportionl. 4. Multiplition Proprty of Equlity 5. ddition Proprty of Equlity 6. istriutiv Proprty of Equlity 7. Sgmnt ddition Postult 8. Sustitution Proprty U1-420 GPS nlyti Gomtry Thr Rsour Wlh Edution

7 UNIT 1 SIMILRITY, ONGRUENE, N PROOFS Lsson 7: Proving Similrity Exmpl 2 Find th lngth of th ltitud, x, of. x Idntify th similr tringls. is right tringl. Th ltitud of is drwn from right to th opposit sid, rting two smllr similr tringls. 2. Us orrsponding sids to writ proportion ontining x. shortr lg of longr lgof shortr lg of longr lgof x x (x)(x) (10)(18) x x Sustitut vlus for h sid. Find th ross produts. Simplify. Tk th positiv squr root of h sid. 3. Summriz your findings. Th lngth of th ltitud, x, of is 6 5 units, or pproximtly 13.4 units. Wlh Edution U1-421 GPS nlyti Gomtry Thr Rsour

8 UNIT 1 SIMILRITY, ONGRUENE, N PROOFS Lsson 7: Proving Similrity Exmpl 3 Find th unknown vlus in th figur f 1. Idntify th similr tringls. is right tringl. Th ltitud of is drwn from right to th opposit sid, rting two smllr similr tringls. 2. Us orrsponding sids to writ proportion ontining. hypotnus of shortrlg of hypotnus of shortr lg of Sustitut vlus for h sid. (8)(6) ()(4.8) Find th ross produts Simplify. 10 Solv for. U1-422 GPS nlyti Gomtry Thr Rsour Wlh Edution

9 UNIT 1 SIMILRITY, ONGRUENE, N PROOFS Lsson 7: Proving Similrity 3. Us orrsponding sids to writ proportion ontining. longr lgof shortrlg of longr lgof shortr lg of ()(6) (8)(4.8) Sustitut vlus for h sid. Find th ross produts. Simplify. 6.4 Solv for. 4. Us orrsponding sids to writ proportion ontining f. longr lgof shortrlg of longr lgof shortr lg of 4.8 f 8 6 Sustitut vlus for h sid. (4.8)(6) (8)( f ) Find th ross produts f Simplify. f 3.6 Solv for f. 5. Summriz your findings. Th lngth of is 10 units. Th lngth of is 6.4 units. Th lngth of f is 3.6 units. Wlh Edution U1-423 GPS nlyti Gomtry Thr Rsour