84 cm 30 cm. 12 in. 7 in. Proof. Proof of Theorem 7-4. Given: #QXY with 6 Prove: * RS * XY
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1 -. Pln Ojetives o use the ie-plitter heorem o use the ringle-ngle- isetor heorem Emples Using the ie-plitter heorem el-worl onnetion Using the ringle-ngle- isetor heorem Mth kgroun - Wht ou ll Lern o use the ie-plitter heorem o use the ringle-ngle- isetor heorem... n Wh o esign sil, s in Emple Proportions in ringles hek kills ou ll Nee for Help he two tringles in eh igrm re similr. Fin the vlue of in eh.. m. mm 4 m 0 m 4 mm mm m mm. 4.. ft in. in. in.. ft ft 9. in. ft Lesson - he ie-plitter heorem represents generliztion of the ringle Misegment heorem from hpter. he onept of similrit mkes possile this generliztion. he ie-plitter heorem pplie to three prllel lines proves the ringle-ngle isetor heorem. More Mth kgroun: p. 4 Using the ie-plitter heorem ou n use similr tringles to prove the following theorem. Ke onepts heorem -4 ie-plitter heorem If line is prllel to one sie of tringle n intersets the other two sies, then it ivies those sies proportionll. Lesson Plnning n esoures ee p. 4E for list of the resoures tht support this lesson. ell inger Prtie hek kills ou ll Nee For intervention, iret stuents to: Using imilr Figures Lesson -: Emple Etr kills, Wor Prolems, Proof Prtie, h. E E E 4 E E E Proof est-king ip Whenever ou see line tht psses through tringle n is prllel to one of the sies, the ie- plitter heorem m ppl. Proof of heorem -4 Given: #X with Prove: X * * X * ttements * X 4 X esons.. Given. & > &, & > &4. If lines re, then orr. ' re >.. #X, #., Postulte X orr. sies of, > re proportionl.. X X +, +. egment ition Postulte X.. ustitute.. X. Propert of Proportions 9 hpter imilrit 9 peil Nees L Hve stuents rw three prllel lines ut two trnsversls. tuents use ruler to mesure the segments interepte on the trnsversls n oserve the re proportionl, n not ongruent. lerning stle: ttile elow Level L eview the properties of proportions, espeill the ross-prout Propert, efore stuents re the proof of heorem -4 n work through the emples. lerning stle: verl
2 uik hek Ke onepts EXMPLE Using the ie-plitter heorem Grie esponse Fin the vlue of. U UV ie-plitter heorem ustitute.? olve for. Use the ie-plitter heorem to fin the vlue of.. +. he following orollr to the ie plitter heorem ss tht prllel lines ivie ll trnsversls proportionll. ou will prove this orollr in Eerise. orollr orollr to heorem -4 If three prllel lines interset two trnsversls, then the segments interepte on the trnsversls re proportionl. U V. / / eh Guie Instrution ehing ip isuss s lss the proof of heorem -4. In prtiulr, hve stuents use lger to show how step follows from step. lterntive Metho n e foun without using the ie-plitter heorem rememering tht sies of similr tringles re proportionl. sk: Wht proportion oul ou write n solve? EXMPLE EXMPLE iversit ome stuents m e unwre of the ifferent shpes of sils for nient n moern ots. If possile, show pitures of ots with ifferent kins of sils. el-worl onnetion ou winsurf with lrge sil in light wins n smll sil in strong wins. uik hek EXMPLE el-worl onnetion il Mking il mkers sometimes use omputer to rete pttern for sil. fter the ut out the pnels of the sil, the sew them together to form the sil. he eges of the pnels in the sil t the right re prllel. Fin the lengths n... ie-plitter heorem. orollr to the ie-plitter heorem (. olve for.. Length is ft n length is. ft. olve for n. ;.. ft ft ft. ft. ft. ft itionl Emples Fin.. he segments joining the sies of trpezoi U re prllel to its ses. Fin n. 9 U M N.,. 0 Lesson - Proportions in ringles 99 vne Lerners L4 Use the ie-plitter heorem to prove tht, if line prllel to one sie of tringle intersets the mipoint of nother sie, then it intersets the mipoint of the thir sie. lerning stle: verl English Lnguge Lerners ELL For Emple, review the terms sil mker n pnels. Mke sure stuents unerstn the piees of the sil re ut n sewn so the ses (eges of the piees re prllel. lerning stle: verl 99
3 Guie Instrution Mth ip fter introuing the ringle- ngle-isetor heorem, remin stuents of the ngle isetor heorem in Lesson -. Hve them eplin how the theorems iffer. ehnolog ip tuents n moel the proof of heorem - using geometr softwre. EXMPLE Error Prevention ome stuents looking t the igrm for this emple m think the n use the proportions from Lesson -4 tht ppl onl to right tringles. sk: Wht must e true to ppl the theorems n orollries from Lesson -4? he tringle must e right tringle with n ltitue to the hpotenuse. Prt Using the ringle-ngle-isetor heorem Proof ou n use the ie-plitter heorem to prove the following reltionship. Ke onepts heorem - ringle-ngle-isetor heorem Prolem olving Hint * E rwing sets up #F for the ie-plitter heorem, s well s ongruent ',,, n 4. If r isets n ngle of tringle, then it ivies the opposite sie into two segments tht re proportionl to the other two sies of the tringle. Proof of heorem - Given: #, isets &. Prove: * E * E rw. Eten to meet t point F. Proof: the ie-plitter heorem, F. the orresponing ngles Postulte, & > &. ine isets &, & > &. the lternte Interior ngles heorem, & > &4. Using the rnsitive Propert of ongruene, ou know tht & > &4. the onverse of the Isoseles ringle heorem, F. ustituting for F,. E F 4 I itionl Emples Fin the vlue of. 4 G K 40 0 esoures il Notetking Guie - L il Notetking Guie - pte Instrution L H uik hek EXMPLE Using the ringle-ngle-isetor heorem lger Fin the vlue of. P P ringle-ngle-isetor heorem ustitute. 4 ross-prout Propert 9. olve for. Fin the vlue of... P losure In, n M isets &. Fin n. 4 M., 4. EXEIE For more eerises, see Etr kill, Wor Prolem, n Proof Prtie. Prtie n Prolem olving Prtie Emple for Help Emple (pge 99 lger olve for hpter imilrit 400
4 Emple (pge 99 Use the figure t the right to omplete eh proportion. 4.. e f e f.. e e f lger olve for e f Prtie ssignment Guie -, -,, -9, , 9, 0,, 40, 44 hllenge 4-4 el-worl Emple (pge 400 ppl our kills onnetion reers ome urn plnners speilize in environmentl issues or histori preservtion. lger olve for Use the figure t the right to omplete eh proportion.. K. KJ JP K KJ J L 9. JP 0. P PM J K KP. KL KM. KP L LW K MW K PM. JP 4. KL KM K JK MW LW Urn esign In Wshington,.., th, th, 9th, n 0th treets re prllel streets tht interset Pennslvni venue n I treet.. How long (to the nerest foot is Pennslvni venue etween 9th treet n th treet? 9 ft. How long (to the nerest foot is Pennslvni venue etween th treet n th treet? ft 0 ft 4 ft. he sies of tringle re m, m, n m long. Fin the lengths, to the nerest tenth, of the segments into whih the isetor of eh ngle ivies the opposite sie..4 m n. m;. m n. m;. m n 9. m. Open-Ene In tringle, the isetor of n ngle ivies the opposite sie into two segments with lengths m n 9 m. How long oul the other two sies of the tringle e? (ution: Mke sure the three sies stisf the ringle Inequlit heorem. nswers m vr. mple: 9 m n. m I t. 0 th 9 th 4 P 00 ft 00 ft th Pennslvni ve. 0 L M W th est Prep 4- Mie eview - Homework uik hek o hek stuents unerstning of ke skills n onepts, go over Eerises, 4,, 4,. onnetion to lger Eerise Use this eerise to hek how well stuents solve omplite proportions. eview solving equtions with vriles on oth sies. Eerises 4, Hve stuents isuss wh the unlele segment hs length - in Eerise 4 n - in Eerise. GP Enrihment Guie Prolem olving etehing pte Prtie Prtie Nme lss te Prtie - Fin the vlues of the vriles for eh regulr hegon. Leve our nswers in simplest ril form.... Eh regulr polgon hs rii n n pothem s shown. Fin the mesure of eh numere ngle Fin the re of eh equilterl tringle, given the rius. Leve our nswers in simplest ril form. 4 q res of egulr Polgons p 9 L4 L L L L Lesson - Proportions in ringles Fin the re of eh regulr polgon to the nerest squre inh in. Person Eution, In. ll rights reserve.. in.. in. in. in.. in. 40
5 reers Eerise 9 urveors use methos of iniret mesurement tht re se on onepts foun in hpter. Enourge stuents to fin out wht mth ourses re neessr for reer in surveing. Eerise 0 his suggests the theorem If n ngle isetor of tringle isets the opposite sie, then the tringle is isoseles. here ws no w for stuents to prove this until the lerne the ringle-ngle-isetor heorem. Eerises ou m nee to review how to solve qurti equtions. onnetion to Environmentl iene Eerise 9 s inustril proution inrese rmtill uring the twentieth entur, worl leers n itizens showe inresing onern out the environment. Oil spills ontminte wter, kill plnts n nimls, n ultimtel ffet the environment fr from the tul spill site. Visul Lerners Eerise 40 ome stuents m fin sptil visuliztion iffiult. Provie moel for them to emine. For guie to solving Eerise 9, see p el-worl for Help onnetion In this glss roof, prllel lines ivie the sies of tringles proportionll. 9. urveing he perimeter of the tringulr lot t the right is 0 m. he surveor s tpe isets n ngle. Fin the lengths n. m; m 0. ritil hinking hrell rws #. he fins tht the isetor of & isets the opposite sie.. keth # n the isetor. ee left. m m. Writing Wht tpe of tringle is #? Eplin our resoning. isoseles; k-l isetor hm. lger olve for , eveloping Proof op n omplete this two-olumn proof of the onverse of the ie-plitter heorem: If line ivies two sies of tringle proportionll, then it is prllel to the thir sie. Given: X Prove: X X ttements esons. X. 9 Given X.. 9 Prop. of Proportions X.. 9 egment. Post. 4. & > &. 9 efleive Prop. of O. #X, # e. 9 M hm.. & > & f. 9 orr. ' of M >re O.. X g. 9 If orr. ' re O, lines n. Proof. Follow the steps elow. Write proof of the orollr to the ie-plitter heorem foun on pge 99. * * * Given: W X Z Prove: WX X * egin rwing W, interseting X t point Z.. ppl the ie-plitter heorem to #W: WZ Z. WX. ppl the ie-plitter heorem to #W: WZ Z. X WX. ustitute to prove the orollr. X etermine whether the re segments re prllel. Eplin eh nswer. ou n use the theorem prove in Eerise 4.. ee mrgin. W X nline Homework Help Visit: PHhool.om We oe: ue GP hpter imilrit 9. es; sine, the segments re n the onverse of the ie- plitter hm.. 4 No; u.. 0 es; sine, the segments re n the onverse of the ie- plitter hm. 40
6 el-worl Multiple hoie hort esponse Etene esponse 9. Oil pills esrie how ou oul use the figure t the right to fin the length of the oil spill iniretl. Wht mesurements n lultions woul ou use? ee mrgin. 40. n ngle isetor of tringle ivies the opposite sie of the tringle into segments m n m long. seon sie of the tringle is. m long. Fin ll possile lengths for the thir sie of the tringle. 4. m or. m Geometr in imensions In the figure t the right, * * * * FG n, GH n, F, FE 4, n G. 4. Fin GE. 4. If EH, fin H.. 4. If FG, fin the perimeter of #E. 9. hllenge Proof 44. One sie of tringle is k times s long s seon sie. he isetor of their ngle uts the thir sie into two segments. Prove tht one of those segments is k times s long s the other. Proof 4. Use the efinition in prt ( to prove the sttements in prts ( n (.. Write efinition for misegment of prllelogrm.. prllelogrm misegment is prllel to two sies of the prllelogrm.. prllelogrm misegment isets the igonls of prllelogrm.. ee k of ook. 4. tte the onverse of the ringle-ngle-isetor heorem. Give onvining s k, so k. rgument tht the onverse is true or ounteremple to prove tht it is flse. ee k of ook. 4. In #, the isetors of &, &, n & ut the opposite sies into lengths n, n, n n, respetivel, lele in this orer ounterlokwise roun #. Fin the perimeter of # for eh of the following..,, 4 4.,, he two segments hve lengths n. ks onnetion ou mesure n oil spill to fin the size of the oom ou ll nee to ontin it. est Prep n 0 0. [] ; n ± 0; n [] orret proportion solve inorretl lesson quiz, PHhool.om, We oe: u-00 9.Mesure, E, n. Use the ie-plitter hm. Write the proport. E n solve for. 4. Use the figure t the right. Wht is? he legs of right tringle hve lengths n 4. he isetor of the right ngle ivies the hpotenuse into two segments. Wht is the length of the shorter segment of the hpotenuse to the nerest tenth? F F.. G..0 H. 9.4 J. 0. Wht is n? how our work. ee left. n + 0. he isetors of n ngle of tringle ivie the opposite sie of the tringle into segments 4 m n m long. seon sie of the tringle is m long.. rw two igrms ou n use to fin the two possile ifferent lengths for the thir sie.. ee k of ook.. Use eh igrm in prt ( to write proportion. olve for eh possile length of the thir sie of the tringle. how our work. 0 + Lesson - Proportions in ringles 40 F E J G H E 4. ssess & eteh Lesson uiz olve for in eh igrm Use the igrm elow for Eerises Fin. 9.. Fin.. Eplin how ou know tht M is not the ngle isetor of &. If it were, M woul equl M. ut. lterntive ssessment Hve eh stuent rw two tringles, onstrut segment prllel to sie on one tringle n n ngle isetor on the other tringle, n then use rulers n lultors to illustrte the ie-plitter n ringle-ngle- isetor heorems. est Prep 4.. M esoures For itionl prtie with vriet of test item formts: tnrize est Prep, p. 4 est-king trtegies, p. 40 est-king trtegies with rnsprenies 40
7 Use this hekpoint uiz to hek stuents unerstning of the skills n onepts of Lessons - through -. esoures Gr & Go hekpoint uiz Mie eview for Help. V. V Lesson -4 Lesson -4 Lesson - efer to the figure to omplete eh proportion.. n m. h h m h n m 4. n. m h n h lger V is retngle. Fin the lengths of the igonls n V.. +, V , V , V , V + 4 V 4 V. Fin the enter of the irle tht ou n irumsrie out eh k. 0. (0, 0 (,. (, (0,. (-, 0 (.,. (, 0 (-, (, (0, - (-, - (-, For k, nme the point of intersetion ssoite with eh set of segments.. the meins 4. the ngle isetors. the ltitues entroi inenter orthoenter hekpoint uiz Lessons - through hpter imilrit etermine whether the tringles re similr. If so, write the similrit sttement. lso, write the postulte or theorem tht proves the re similr.. X. G H k M kxz; M Post. W Z J kw M khjg; M hm. lger he polgons re similr. Fin the vlue of eh vrile.. w 4. m " ; w 4. lger Fin the vlue of eh vrile "
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