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1 GeometryFall2016 Name:Period: Unit2–Congruence Designedtobeusedtogetherwith MathematicsVisionProjectPacket(Year2)Module5 1 2 GeometryHomeworkGuideUnit2 Period_____Name______________________ Date LearningTarget/Standard Homework EntryTask EntryTaskTotal: ImportantDates: 2 3 Name_______________________________________________ FindingCongruentTriangles 1) Alltrianglesaremadeupofthreesidesandhavethreeinteriorangles.Usingaprotractor,measureallthree anglesineachofthetrianglesbelowmeasuredtothenearestdegree: a. ∆"#$ m∠" = m∠# = m∠$ = A C B b. ∆'() m∠' = m∠( = m∠) = c. ∆*+, m∠* = m∠+ = m∠, = P Q R d. Makeaconjectureaboutthemeasuresoftheanglesfoundinatriangle.Whatdothemeasuresofthe anglesintrianglesallseemtohaveincommon?(Wewillprovethisconjecturetobetruelater.) 3 4 2) Giventhattherearesixpartstoeverytriangle(threesidesandthreeangles)itturnsoutthatwhenwecompare anytwoofthemtoseeiftheyarethesametriangle“congruent”,weonlyneedtocompareanycombinationof threeofthoseparts.Listallthewaysthatwecanchoosethreepartsofatriangletocompare: A S S A A S 3) Whenwechoosethreepartsofatriangleandcomparethemtotheircorrespondingpartsofadifferenttriangle, sometimeswefindthetwotrianglesarecongruentandsometimeswefindtheyarenotcongruent.Inthis activity,youwillchooseanythreepartsofatriangleandinvestigatewhetherthosethreepartscreateexactly onetriangleorifthosethreepartscanmakemorethanonedifferenttriangle.Namethepartstakenandcircleif onlyonetrianglecanbeformedorifmorethanonetrianglecanbemade: 3PartsTaken NumberofPossibleTriangles ExactlyOneTrianglecanbeformed MorethanoneTrianglecanbeformed ExactlyOneTrianglecanbeformed MorethanoneTrianglecanbeformed ExactlyOneTrianglecanbeformed MorethanoneTrianglecanbeformed ExactlyOneTrianglecanbeformed MorethanoneTrianglecanbeformed ExactlyOneTrianglecanbeformed MorethanoneTrianglecanbeformed ExactlyOneTrianglecanbeformed MorethanoneTrianglecanbeformed ExactlyOneTrianglecanbeformed MorethanoneTrianglecanbeformed 4) Ifonlyonetrianglecanbeformedgiventhreepartsofatriangle,thenthatissufficientevidenceneededto provetwotrianglesarecongruent.Ithethreepartsofonetrianglearecongruenttothreecorrespondingparts ofanothertriangle,thetwotrianglesarecongruent.Showthetickmarksthatwouldindicatethetrianglesare congruentinfivedifferentways: 4 5 5) Indicateifthetwotrianglesarecongruent≅ornotcongruent≅giventhemarks.Iftheyarecongruent,justify thisbyindicatingthethreecorrespondingpartsaslistedonthepreviouspage: 5 6 6 7 CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 7. 5 Congruent Triangles to the Rescue A Practice Understanding Task Part1 ZacandSioneareexploringisoscelestriangles—trianglesinwhichtwosidesarecongruent: Zac:Ithinkeveryisoscelestrianglehasalineofsymmetrythatpassesthroughthevertex pointoftheanglemadeupbythetwocongruentsides,andthemidpointofthethirdside. Sione:That’saprettybigclaim—tosayyouknowsomethingabouteveryisoscelestriangle. Maybeyoujusthaven’tthoughtabouttheonesforwhichitisn’ttrue. Zac:ButI’vefoldedlotsofisoscelestrianglesinhalf,anditalwaysseemstowork. Sione:Lotsofisoscelestrianglesarenotallisoscelestriangles,soI’mstillnotsure. 1. WhatdoyouthinkaboutZac’sclaim?Doyouthinkeveryisoscelestrianglehasalineof symmetry?Ifso,whatconvincesyouthisistrue?Ifnot,whatconcernsdoyouhaveabout hisstatement? 2. WhatelsewouldZacneedtoknowaboutthecreaselinethroughinordertoknowthatitisa lineofsymmetry?(Hint:Thinkaboutthedefinitionofalineofreflection.) 3. SionethinksZac’s“creaseline”(thelineformedbyfoldingtheisoscelestriangleinhalf) createstwocongruenttrianglesinsidetheisoscelestriangle.Whichcriteria—ASA,SASor SSS—couldheusetosupportthisclaim?Describethesidesand/oranglesyouthinkare congruent,andexplainhowyouknowtheyarecongruent. 4. Ifthetwotrianglescreatedbyfoldinganisoscelestriangleinhalfarecongruent,whatdoes thatimplyaboutthe“baseangles”ofanisoscelestriangle(thetwoanglesthatarenot formedbythetwocongruentsides)? Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 7 https://flic.kr/p/3GZScG SECONDARY MATH I // MODULE 7 CC BY Anders Sandberg 24 25 8 SECONDARY MATH I // MODULE 7 CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 5. Ifthetwotrianglescreatedbyfoldinganisoscelestriangleinhalfarecongruent,whatdoes thatimplyaboutthe“creaseline”?(Youmightbeabletomakeacoupleofclaimsaboutthis line—oneclaimcomesfromfocusingonthelinewhereitmeetsthethird,non-congruent sideofthetriangle;asecondclaimcomesfromfocusingonwherethelineintersectsthe vertexangleformedbythetwocongruentsides.) Part2 LikeZac,youhavedonesomeexperimentingwithlinesofsymmetry,aswellasrotational symmetry.InthetasksSymmetriesofQuadrilateralsandQuadrilaterals—BeyondDefinitionyou madesomeobservationsaboutsides,angles,anddiagonalsofvarioustypesofquadrilateralsbased onyourexperimentsandknowledgeabouttransformations.Manyoftheseobservationscanbe furtherjustifiedbasedonlookingforcongruenttrianglesandtheircorrespondingparts,justasZac andSionedidintheirworkwithisoscelestriangles. Pickoneofthefollowingquadrilateralstoexplore: • Arectangleisaquadrilateralthatcontainsfourrightangles. • Arhombusisaquadrilateralinwhichallsidesarecongruent. • Asquareisbotharectangleandarhombus,thatis,itcontainsfourrightanglesand allsidesarecongruent 1. Drawanexampleofyourselectedquadrilateral,withitsdiagonals.Labeltheverticesofthe quadrilateralA,B,C,andD,andlabelthepointofintersectionofthetwodiagonalsaspointN. 2. Basedon(1)yourdrawing,(2)thegivendefinitionofyourquadrilateral,and(3)information aboutsidesandanglesthatyoucangatherbasedonlinesofreflectionandrotational symmetry,listasmanypairsofcongruenttrianglesasyoucanfind. 3. Foreachpairofcongruenttrianglesyoulist,statethecriteriayouused—ASA,SASorSSS—to determinethatthetwotrianglesarecongruent,andexplainhowyouknowthattheangles and/orsidesrequiredbythecriteriaarecongruent(seethefollowingchart). Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 8 26 9 SECONDARY MATH I // MODULE 7 CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 Congruent Triangles CriteriaUsed (ASA,SAS,SSS) IfIsayΔRST≅ΔXYZ basedonSSS HowIknowthesidesand/orangles requiredbythecriteriaarecongruent thenIneedtoexplain: • howIknowthat RS ≅ XY ,and • howIknowthat ST ≅ YZ ,and • howIknowthat TR ≅ ZX soIcanuseSSScriteriatosayΔRST≅ΔXYZ 4. Nowthatyouhaveidentifiedsomecongruenttrianglesinyourdiagram,canyouusethe congruenttrianglestojustifysomethingelseaboutthequadrilateral,suchas: • thediagonalsbisecteachother • thediagonalsarecongruent • thediagonalsareperpendiculartoeachother • thediagonalsbisecttheanglesofthequadrilateral Pickoneofthebulletedstatementsyouthinkistrueaboutyourquadrilateralandtryto writeanargumentthatwouldconvinceZacandSionethatthestatementistrue. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 9 28 10 SECONDARY MATH I // MODULE 7 7.5 CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 Afterworkingwiththeseequationsandseeingthetransformationsonthecoordinategraphitisgood timingtoconsidersimilarworkwithtables. 6.Matchthetableofvaluesbelowwiththeproperfunctionrule. I II x -1 0 1 2 f(x) 16 14 12 10 III x -1 0 1 2 A.! ! = −! ! − ! + ! B.! ! = −! ! − ! + !" C.! ! = −! ! − ! + ! f(x) 14 12 10 8 IV x -1 0 1 2 f(x) 12 10 8 6 V x -1 0 1 2 D.! ! = −! ! + ! + ! E.! ! = −! ! + ! + !" f(x) 10 8 6 4 x -1 0 1 2 SET Topic:UseTriangleCongruenceCriteriatojustifyconjectures. Ineachproblembelowtherearesometruestatementslisted.Fromthesestatementsa conjecture(aguess)aboutwhatmightbetruehasbeenmade.Usingthegivenstatementsand conjecturestatementcreateanargumentthatjustifiestheconjecture. 7.Truestatements: Conjecture: ∠A ≅ ∠C PointMisthemidpointof!" ∠!"# ≅ ∠!"# a.Istheconjecturecorrect? !" ≅ !" b.Argumenttoproveyouareright: Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 10 f(x) 8 6 4 2 29 11 SECONDARY MATH I // MODULE 7 7.5 CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 8.Truestatements ∠ !"# ≅ ∠ !"# !" ≅ !" Conjecture:!"bisects∠ !"# a.Istheconjecturecorrect? b.Argumenttoproveyouareright: Conjecture:∆ !"# ≅ ∆!"# a.Istheconjecturecorrect? b.Argumenttoproveyouareright: 9.Truestatements ∆ !"#isa180° rotationof∆ !"# GO Topic:Constructionswithcompassandstraightedge. 10.Whydoweuseageometriccompasswhendoingconstructionsingeometry? Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 11 12 ExtraPractice/Homework 12 13 Kuta Software - Infinite Geometry Name___________________________________ Parallel Lines and Transversals Date________________ Period____ Identify each pair of angles as corresponding, alternate interior, alternate exterior, or consecutive interior. 1) 2) y y x x 3) 4) y y x x 5) 6) y y x x 7) 8) y x x y ©e Z2t0h1Z1k qKUuNtraS ZSZoHfRtqwnacr6e5 eLKLSCZ.C U 3ASl1lL Qr3iRguhNt2sE srIeYs0eIrXvYePd2.7 Z xMkakdJeO lwaiItWh9 tIxnvf9iCnxiGtnes LGKecoTmTeZthrxyG.0 13 -1- Worksheet by Kuta Software LLC 9) 14 10) y x x y Find the measure of each angle indicated. 11) 12) ? 84° ? 110° 13) 14) ? 100° 111° ? 15) 16) ? 125° ? 17) 47° 18) 53° ? 113° ? ©i L2c0j1e12 wKcuXtwaN NSmoFfftDwxaTr1eS OLwLXCZ.4 b GAKlzlx 7rHi1gVhntisE 5r1eJsge5rwvteOd3.o Z TMTaOd0eF DwQihtohB TInnAfViYnRi5tveR 4G1eTo7mUeitOrGyL.k 14 -2- Worksheet by Kuta Software LLC 15 Solve for x. 19) 20) 75° 11 x − 2 21 x + 6 21) 22) x + 139 60° 132° 8x − 4 23) 24) −1 + 14 x 23 x − 5 12 x + 17 21 x + 5 Find the measure of the angle indicated in bold. 25) 26) x + 96 20 x + 5 x + 96 24 x − 1 27) 28) 6x x + 109 x + 89 5 x + 10 ©F z2O0A1H1N EKTuttpaK YS1o3fQt4wIahryeN gL1LFCl.s W tA1lql4 krriwgbhztjsh krNe6smeVrevVeBdI.1 Q yMNa9djeL ew4iPtPhT kINnAf2iQnjiQtmeq eGjeiopmye8tVrJyM.o 15 -3- Worksheet by Kuta Software LLC 16 Kuta Software - Infinite Geometry Name___________________________________ Angles in a Triangle Date________________ Period____ Find the measure of each angle indicated. 1) 2) ? 65° 40° 57° ? 3) 4) 85° 20° 50° ? 130° ? 5) 6) 137° ? 102° 100° ? 35° 7) 8) 155° 60° ? 30° ? 20° ©J K2L0H181m eK4uNtUaN 9STo5fBtMwXa2ree9 sLYLRCP.H g CAHlGlA PrCiAgRhatqsv rrxevsve3rRvceOd8.K 5 CMoaBd9e8 5wAiAtghM vIqnEf6i7nNi8tre7 QGYeXoEm9eatyrQyf.Q 16 -1- Worksheet by Kuta Software LLC 9) 17 10) ? 39° 35° 40° 20° 65° 50° 60° 11) ? 12) 86° ? 27° 84° ? 35° 23° 36° 13) 14) 156° ? 155° 115° 35° 20° 85° ? 15) 16) 45° 45° 68° 68° 79° 100° ? 75° 60° ? ©s m2a0q1m1S MKWuGtfad 6SmozfotkwEaWrDe1 GLbLfCl.I h yAKlklM kr4ingCh7tes7 BrqeVsIe9rXv4e7dC.r S pMPaadJed WwAiXtjhN GIsndfGiznFiXtoeq 8GlemoymfeUtgr1yh.c 17 -2- Worksheet by Kuta Software LLC 18 Solve for x. 17) 18) 55° x + 74 8x + 2 54° 70° 19) 60° 20) 64° 27° x + 51 97 + x 60° 80° Find the measure of angle A. 21) 22) x + 59 x + 37 x + 67 A 84° x + 51 A 23) 24) A A 3x − 6 130° x + 23 8x + 4 ©k 22X0n1v1k 3KjurtSad ySMonfUtBwZagrVe3 hLvLCCh.F N 4AJlilo greiUgNhVtJsT Xr8eCsBenrGvAezdn.Y V 6MlaEdhe2 AwwiDtUhX fIUnEf1iknUittWe8 wG7ehosmUeotUrSyh.B 80° 4 x + 17 18 -3- Worksheet by Kuta Software LLC 19 Kuta Software - Infinite Geometry Name___________________________________ ASA and AAS Congruence Date________________ Period____ State if the two triangles are congruent. If they are, state how you know. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) ©1 02h0w1b26 8Kduftrak jSXo0fMthwUairseU kLXL0Cs.H B QAVl7la grNiig1hqtws5 kr5efsteorsvHeQdN.A l nMEaPdQeG lwdirtihN LIynzftiSnXirtAen OGyeHofm8eqt9rQy0.X 19 -1- Worksheet by Kuta Software LLC 21 Kuta Software - Infinite Geometry Name___________________________________ SSS and SAS Congruence Date________________ Period____ State if the two triangles are congruent. If they are, state how you know. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) ©f M2x0F1S1l YK0uMtwav oSPopfYtpwNaSruez qLULSC7.5 p jA5ljls ordi2gDhctisS trseMsqeVrBvCeFdw.c f IMMandSeQ Gw8i3tShv uIonjf2iRnqiZtmeY vGMeLogmQeCtZrPyl.X 21 -1- Worksheet by Kuta Software LLC 23 Kuta Software - Infinite Geometry Name___________________________________ Congruence and Triangles Date________________ Period____ Complete each congruence statement by naming the corresponding angle or side. 1) ∆DEF ≅ ∆KJI 2) ∆BAC ≅ ∆LMN E K I M A C B F D N J ∠A ≅ FD ≅ ? 3) ∆TUV ≅ ∆GFE ? 4) ∆WVU ≅ ∆GHI U V W I G F G U E T ∠W ≅ V ∠U ≅ L H ? ? 5) ∆ZXY ≅ ∆ZXC 6) ∆DEF ≅ ∆DSR Y E D Z X F R S C ∠Y ∠F ≅ ≅? ? Write a statement that indicates that the triangles in each pair are congruent. 8) 7) T J I D R H I K S ©Q g20031Z1Q bKyuEtPal jSwoIfntWwvarrZeJ rLkLzCE.F a cAmlylB grCiogIhntos6 vrLessBe2rFvNeNdN.c U RM8aedfeF ewCittxhh QIinGftiFnuirtseV yGJejoKm0e1tFrGyU.k C B G 23 -1- Worksheet by Kuta Software LLC 24 9) 10) R S I P Q T R D 11) 12) E C D U C X W V D S T E Mark the angles and sides of each pair of triangles to indicate that they are congruent. 14) ∆GFE ≅ ∆LKM 13) ∆BDC ≅ ∆MLK D L K M G E B C L 15) ∆MKL ≅ ∆STL F M K 16) ∆HIJ ≅ ∆JTS K T I S M 17) ∆CDB ≅ ∆CDL S H T L J 18) ∆JIK ≅ ∆JCD C K I L J B D ©Q T2m0L111J oKMuft0a8 YSFovfut1wAaprzeH YLlL4CN.a 7 qAjlHl8 srMi1g5h7tpsE YrleisYeDrIvyeydr.C w 4Mfaadmem pwjiptQhE ZIOnGfSi0nuiqtceu sGde1oBmVeRtbrHyo.q D C 24 -2- Worksheet by Kuta Software LLC 25 Kuta Software - Infinite Geometry Name___________________________________ Isosceles and Equilateral Triangles Date________________ Period____ Find the value of x. 1) 2) 7 x x 6 3) 4) 6 x 4 5) x 6) 40° x x 7) 75° 8) x 75° x 54° 9) 10) 65° x 28° x ©W R2R0Y1p1Y aKFuCtTan 5SnogfStww2asr1eA xL4LeCs.A b bAzlEl6 orwijgdhOt6sm OrnezsqeqrbvMeadr.U b IMhaldVeb BwCiVthhG zIanifZi2n3iutYeu oGye1ormoeRtorvyp.q 25 -1- Worksheet by Kuta Software LLC 11) 26 12) x 120° 148° x 13) 14) 8 −1 + x 12 15) m∠2 = x + 94 2 x − 12 16) m∠2 = 4 x − 2 2 2 68° 47° 17) m∠2 = 12 x + 4 18) m∠2 = 13 x + 3 2 118° 146° 2 ©K r250b1a19 4KmuBtraE tS9o7fotCwSanrRed yLaL1CW.G k sA3l7lt UrBiXghhytvsv rreeAsbesrzvPeGdh.X k nMfaFdrej vwei4tthw oIhnRfri8n5iwteeL uG5exo8mie6trrqyh.a 26 -2- Worksheet by Kuta Software LLC 29 29 30 30 31 31 32 32 33 33 34 34