-. ln Objectives o recognize congruent figures nd their corresponding prts xmples ming ongruent rts 2 el-world onnection inding ongruent ringles roving ringles ongruent - Wht ou ll ern o recognize congruent figures nd their corresponding prts...nd Why o use corresponding prts of congruent shpes in the pce huttle, s in xmple 2 ongruent igures x 2 heck kills ou ll eed O for elp lgebr olve ech eqution.. x + 6 = 2 9 2. x + 7 + =. x = 0 08. x + 0 = 2x 0. or the tringle t the right, use the ringle ngle-um heorem to find the vlue of y. 0 ew Vocbulry congruent polygons lgebr eview, pge 0 0 y th ckground ongruent polygons hve one-to-one correspondence of equlity between ll of their corresponding prts. ny proofs concerning polygons nd circles rely on proving tringles congruent. ore th ckground: p. 96 ongruent igures ongruent figures hve the sme size nd shpe. When two figures re congruent, you cn move one so tht it fits exctly on the other one. hree wys to mke such move slide, flip, nd turn re shown below. ou will lern much more bout slides, flips, nd turns in hpter 9. lide urn esson lnning nd esources lip ee p. 96 for list of the resources tht support this lesson. oweroint ell inger rctice heck kills ou ll eed or intervention, direct students to: olving iner qutions lgebr eview, pge 0 uick heck ongruent polygons hve congruent corresponding prts their mtching sides nd ngles. tching vertices re corresponding vertices. When you nme congruent polygons, lwys list corresponding vertices in the sme order. ming ongruent rts # > #. ist the congruent corresponding prts. ides: > > > ngles: & > & & > & & > & #W > #V. ist the congruent corresponding prts. Use three letters to nme ech ngle. lw O lv; lw O lv; lw O lv; W O ; W O V ; O V corresponds to. corresponds to. corresponds to. 98 hpter ongruent ringles 98 pecil eeds how tht there is more thn one wy to write congruence correspondence. or xmple, the congruent tringles cn be expressed in six wys:,,,,, nd. lerning style: visul elow evel 2 ve students nme in six different wys, nd explin why is correct, in the lst congruence sttement of xmple, nd the other five wys re incorrect. lerning style: visul
uick heck uick heck 2 el-world onnection pcecrft he fins of the pce huttle suggest congruent pentgons. ind m&. In the congruent pentgons, corresponds to, so you know tht & > &. ou cn find m& by first finding m&. Use the olygon ngle-um heorem. It tells you tht the sum of the mesures of the ngles of pentgon is ( - 2)80, or 0. m& + m& + m& + m& + m& = 0 olygon ngle-um heorem 88 + 90 + 90 + 2 + m& = 0 ubstitute. 00 + m& = 0 implify. m& = 0 ubtrct 00 from ech side. m& = m&, so m& = 0. 2 It is given tht #W > #V. If m& =, wht is m&? xplin. ml ; corr. ' re O. wo tringles re congruent when they hve three pirs of congruent corresponding sides nd three pirs of congruent corresponding ngles. inding ongruent ringles ecide whether the tringles re congruent. ustify your nswer. > iven > > & > & iven & > & ll right ngles re congruent. & > & Verticl ngles re congruent. # > # by the definition of congruent tringles. n you conclude # > #? ustify your nswer. o; corr. sides re not necessrily O. 88 he next theorem follows from the ringle ngle-um heorem. In xercise, you will show why this theorem is true. 2 W 2. ech uided Instruction eching ip iscuss how the sttement llows you to list corresponding prts correctly without referring to the digrm. th ip emind students tht & cn be nmed & nd tht is the sme segment s. s students exmine the steps, sk: Why re some ngles nmed with one letter nd other ngles with three letters? One letter is used when it is the vertex of only one ngle. uditory erners When you present heorem -, sk students to suggest ides for ln for roof. ed discussion of students ides. oint out tht step 2 of the proof uses the eflexive roperty, which is used extensively in geometry. sk: When you look t the digrm, why would you use the eflexive roperty? to show tht the third sides of the tringles re congruent ey oncepts heorem - If two ngles of one tringle re congruent to two ngles of nother tringle, then the third ngles re congruent. & > & esson - ongruent igures 99 dvnced erners ve students write two-column or flow proof of heorem -. lerning style: verbl nglish nguge erners elp students recognize tht corresponding prts refers to mtching sides, mtching ngles, nd mtching vertices. mphsize tht congruent figures re nmed by listing vertices in mtching order. lerning style: verbl 99
oweroint dditionl xmples. ist the congruent corresponding prts. l Ol, l Ol, l Ol, O, O, O 2, m& = 67, nd m& = 8. ind m&. 6 xplin why is not congruent to in xmple. orresponding sides re not congruent. iven:,,, nd re right tringles. rove:. roof. l O l; l O l (iven) b. l O l (Vert. ' re O.) c. O ; O ; O (iven) d. k O k (ef. of O >) uick heck xmple (pge 98) xmple shows typicl setup iven, rove, nd digrm tht requires proof from you. he form of proof you use is generlly mtter of preference. roving ringles ongruent iven: >, >, / > /, / > / rove: n > n ttements esons. >, >. iven 2. > 2. eflexive roperty of >. & > &, & > &. iven. & > &. heorem -. # > #. efinition of > tringles iven: & > &, & > &, >, >, > rove: # > # ee left. I or more exercises, see xtr kill, Word roblem, nd roof rctice. rctice nd roblem olving O rctice by xmple for elp. uilding uilders use the ing ost truss, below left, for the top of simple structure. In this truss, # > #. ist the congruent corresponding prts. l O l; l O l; l O l; O ; O ; O. O (iven) b. O (iven) c. O (eflexive rop. of O) d. l Ol (iven) e. l Ol (ight ngles re O.) f. l Ol (h. -) g. k Ok (ef. of O tringles) esources ily otetking uide - ily otetking uide - dpted Instruction el-world onnection xposed bems show the congruent tringles used in udor rchitecture. I 2. he ttic rme truss, bove right, provides open spce in the center for storge. In this truss, # > #I. ist the congruent corresponding prts. l O li; l O li; l O li; O ; O I ; O I k Ok. omplete the congruence sttements.. > 9. > 9. > 9 6. & > 9 l 7. & > 9 l 8. & > 9 l 9. # > 9 0. # > 9 k k. # > 9 2. # > 9 k k. he lst piece of the jigsw puzzle must be put into plce. me the corners tht correspond to corners,,, nd.,,, losure uppose tht two pentgons re congruent. ow mny pirs of congruent corresponding prts re there? xplin. t lest 0 pirs; pirs of congruent ngles nd pirs of congruent sides O O I. ist ech of the following.. four pirs of congruent sides. four pirs of congruent ngles O O I ; O O I ; O ; O l O l; lo O li; l O l; l O l 200 hpter ongruent ringles 200
xmple 2 (pge 99) In the two lifegurd chirs, O I. ind the mesure of the ngle or the length of the side.. rctice xmple (pge 99) 6 in. 0 0 in. In xercises 2 27, cn you conclude the figures re congruent? ustify ech nswer. 2. # nd #U ee left. 2. # nd #UV 6. in. 7. I in. 8. & 0 9. & 77 in. 20. 6 in. 2. in. in. 22. & 7 2. &I 0 77 7 I 2. yes; l O lu, l O lu (iven) l O lu V 7 (If two ' of k re O U 8 to two ' of nother k, 6 the third ' re re O.) 7 U O U, O U o; the corr. sides re not O. (iven) O (eflexive rop. of O) 26. # nd # o; the corr. 27. nd o k O ku by the def. of O >. sides re not necessrily O. 20 60 es; ll corr. sides nd ' re O. ssignment uide - hllenge 6-8 est rep 9-2 ixed eview -9 omework uick heck o check students understnding of key skills nd concepts, go over xercises 2, 28, 0, 2,. rror revention! xercise 2 tudents my think tht & corresponds to & becuse they re in the sme reltive positions. ncourge them to use the congruence sttement to mrk congruent ngles nd sides on copies of the tringles. xercise 29 ncourge students to copy the figures on their own pper nd mrk the congruent sides nd ngles. xmple (pge 200) roof 28. iven: 6, & > &, >, > rove: # > # ee mrgin. O pply our kills for elp o review the ringle ngle-um heorem, go to esson -. x 2 lesson quiz, chool.com, Web ode: u-00 28.. O, O re given. O by the efl. rop. l Ol is given, nd by the lt. l hm., l Ol 29. ultiple hoice If # > #, which of the following must be correct congruence sttement? > & > & > & > & lgebr ind the vlues of the vribles. 0. x ;. x t 2 in. 2t in. nd l Ol. o k Ol by the def. of O k. 6x 0 esson - ongruent igures 20 nrichment uided roblem olving eteching dpted rctice rctice erson duction, Inc. ll rights reserved. me lss te rctice - ch pir of polygons is congruent. ind the mesures of the numbered ngles.. 2.. 0 k Ok. ist ech of the following.. three pirs of congruent sides. three pirs of congruent ngles W O. ist ech of the following. 6. four pirs of congruent sides W 7. four pirs of congruent ngles tte whether the pirs of figures re congruent. xplin. 8. nd I 9. nd V 2 I 20 9 9 I V 0. eveloping roof Use the informtion given in the digrm. ive reson tht ech sttement is true.. b. c. d.,, e. W ongruent igures nd orresponding rts 0 V U 0 7 6 8 2 20
. ssess & etech oweroint esson uiz In xercises nd 2, qudrilterl W O qudrilterl O.. ist the congruent corresponding prts. W O O, O O, O, W O ; lw Ol, l OlO, l Ol, l Ol 2. m&o = m& = 90 nd m& = 6. ind m&.. Write sttement of tringle congruence. mple: k Ok. Write sttement of tringle congruence. mple: k Ok. xplin your resoning in xercise bove. mple: wo pirs of O corresponding sides nd two pirs of O corresponding ngles re given. l Olbecuse ll right ngles re congruent. O by the eflexive roperty of O. k Ok by the definition of O tringles. O nline omework elp Visit: chool.com Web ode: ue-00 6. nswers my vry. mple: It is importnt tht O O for the ptch to completely fill the hole.. nswers my vry. mple: k O k: O ; O ; O ; l O l; l O l; l O l xercise 2 hllenge lgebr k Ok. ind the mesures of the given ngles or the lengths of the given sides. ml ml 20 2. m& = x + 0, m& = 2x ml ml 2. m& = y, m& = 2. = z + 2, = z + 6. = 7 +, = + 9 8 6. rquet loor xplin why it is 9 importnt tht > O. ee left. 7. ports rds he 22 crds in rcy s sports crd collection re rectngles of three different sizes. O escribe how rcy could quickly sort the crds. nswers my vry. xercise 6 mple: rcy should rrnge them in pile nd pull out the ones of like sizes. Write congruence sttement for ech pir of tringles. 8. 9. 0. x 2 k O k is the midpoint of.. omplete in two different k O k wys: k O k; # > 9 k O k 2. Writing ie-cst toys re populr collector s item. xplin why the two die-cst toys tht erl is studying t the left hve congruent shpes. ee mrgin.. Open-nded Write congruence sttement for two tringles. ist the congruent sides nd ngles. ee bove left. roof. iven: 6, >. rove heorem -. >, bisects. iven: & > &, & > & rove: # > # rove: & > &. ee mrgin. oordinte eometry Vertices of # re ( 2, ), ( 2, ), nd (, ). 6. # > #. ind,, nd. ; ; 7. If nd hve coordintes (, -) nd (6, -), how mny pirs of coordintes re possible for? ind one such pir. 2; either (, ) or (, 7) bisects. k O k 8.. ow mny qudrilterls (convex nd concve) with different shpes or sizes cn you mke on three-by-three geobord? One is shown t the right. b. ow mny qudrilterls of ech type re there? ee mrgin p. 20. 202 hpter ongruent ringles 2. nswers my vry. mple: he die is mold tht is used to mke items tht re ll the sme size. 202. O, O (iven), O (def. of bisect), l Ol (Vert. l re O.), l Ol (lt. Int. l hm.), l Ol (If 2 l of k re O to 2 l of nother k, the third l re O.) o k Ok by the def. of O k.. l Ol, l Ol (iven), ml ± ml ± ml 80, ml ± ml ± ml 80 (k-l um hm.), ml ± ml ± ml ml ± ml ± ml (ubst. rop.), ml ± ml ± ml ml ± ml ± ml (ubst. rop.), ml ml (ubtr.)
est rep ridded esponse ixed eview Use the digrms t the right for xercises 9. >. 9. Wht is the vlue of?. 0. Wht is the vlue of x?.2. Wht is the perimeter of? 0 2. # > #, m& = 66, nd m& = 2. Wht is m&? 72 8 6 0 x 7 lterntive ssessment rw qudrilterl I on the bord. fter students copy I, hve them drw nd mrk qudrilterl U congruent to I nd write congruence sttements for the figures nd ll the congruent corresponding prts. I O for elp esson -8 onstructions or xercises nd, construct the geometric figure.. ee mrgin.. squre 7. rectngle whose length is twice its width esson - esson 2-. ind m& in the figure t the right. 00 Use the given property to complete ech sttement. xercise 6. ymmetric roperty of qulity 7. eflexive roperty of ongruence If =, then 9. & > 9 l 8. ddition roperty of qulity 9. rnsitive roperty of ongruence If m& - = 8, then m& =9. If > nd >, then 9. 2 O est rep sheet of blnk grids is vilble in the est-king trtegies with rnsprencies booklet. ive this sheet to students for prctice with filling in the grids. esources or dditionl prctice with vriety of test item formts: tndrdized est rep, p. 2 est-king trtegies, p. 28 est-king trtegies with rnsprencies eometry t Work ie sting wo centuries go, people mnufctured rticles by hnd. ch rticle produced ws slightly different from every other. In 800, inventor li Whitney recognized tht he could speed up mnufcturing by using congruent prts. Whitney mde die, or mold, for ech prt of musket he ws producing for the U.. rmy. his llowed workers to rpidly cst the prts nd ssemble them into stndrd-sized muskets. It ushered in the er of mss production. ody, die mkers re highly skilled industril workers who shpe dies out of metl, plstic, rubber, nd other mterils. chines crete nd ssemble the congruent die-cst prts into stndrd-sized objects, like the die-cst toy crs t the left. Other workers supply finl inspection nd skilled hnd finishing.. nswers my vry. mple: chool.com or: Informtion bout die csting Web ode: ub-20 esson - ongruent igures 20. 8. b. 8 8 8 8 8 8 20