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Unit3Congruence&Proofs
Lesson1:IntroductiontoTriangleProofsOpeningExerciseUsingyourknowledgeofangleandsegmentrelationshipsfromUnit1,fillinthefollowing:Definition/Property/Theorem Diagram/KeyWords Statement
DefinitionofRightAngle
DefinitionofAngleBisector
DefinitionofSegmentBisector
DefinitionofPerpendicular
DefinitionofMidpoint
AnglesonaLine
AnglesataPoint
AnglesSumofaTriangle
VerticalAngles
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Example1Wearenowgoingtotakethisknowledgeandseehowwecanapplyittoaproof.Ineachofthefollowingyouaregiveninformation.Youmustinterpretwhatthismeansbyfirstmarkingthediagramandthenwritingitinproofform. a. Given:Disthemidpointof AC
Statements Reasons
1. DisthemidpointofAC 1. Given
2. 2. b. Given: BD bisects AC
Statements Reasons
1. BD bisects AC 1. Given
2. 2. c. Given: BD bisects∠ABC
Statements Reasons
1. BD bisects∠ABC 1. Given
2. 2. d. Given: BD ⊥ AC
Statements Reasons
1. BD ⊥ AC 1. Given
2. 2. 3. 3.
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Example2Listedbelowareotherusefulpropertieswe’vediscussedthatwillbeusedinproofs.
Property/Postulate InWords Statement
AdditionPostulate Equals added to equals are equal.
SubtractionPostulate Equals subtracted from equals are equal.
MultiplicationPostulate Equals multiplied by equals are equal.
DivisionPostulate Equals divided by equals are equal.
PartitionPostulate The whole is equal To the sum of its parts.
SubstitutionA quantity may be
substituted for an equal quantity.
Reflexive Anything is equal to itself
Thetwomostimportantpropertiesaboutparallellinescutbyatransversal:1.2.
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HomeworkGiventhefollowinginformation,markthediagramandthenstateyourmarkingsinproofform. 1. Given: AC bisects∠BCD
Statements Reasons
1. AC bisects∠BCD 1.Given
2. 2. 2. Given:Eisthemidpointof AB
Statements Reasons
1. Eisthemidpointof AB 1.Given
2. 2. 3. Given:
Statements Reasons
1. 1.Given
2. 2. 3. 3. 4. Given: CE bisects BD
Statements Reasons
1. CE bisects BD 1.Given
2. 2.
CD ⊥ AB
CD ⊥ AB
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B'
C'
A'
A
B
C
B"
C"
A
B
C
B"
C"
A
B
C
B'
C'
A'
A
B
C
B'''
A
B
C
Lesson2:CongruenceCriteriaforTriangles-SASOpeningExerciseInUnit2wedefinedcongruenttomeanthereexistsacompositionofbasicrigidmotionsoftheplanethatmapsonefiguretotheother.Inordertoprovetrianglesarecongruent,wedonotneedtoprovealloftheircorrespondingpartsarecongruent.Insteadwewilllookatcriteriathatrefertofewerpartsthatwillguaranteecongruence.Wewillstartwith:Side-Angle-SideTriangleCongruenceCriteria(SAS)
• TwopairsofsidesandtheincludedanglearecongruentUsingthesedistincttriangles,wecanseethereisacompositionofrigidmotionsthatwillmapΔA 'B 'C ' toΔABC .Step1:Translation Step2:Rotation Step3:Reflection
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Example1InordertouseSAStoprovethefollowingtrianglescongruent,drawinthemissinglabels: a
b. Twopropertiestolookforwhendoingtriangleproofs: VerticalAngles ReflexiveProperty (CommonSide)
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Examples2. Given:∠LNM ≅ ∠LNO, MN ≅ON
a. Prove:ΔLMN ≅ ΔLON
b. Describetherigidmotion(s)thatwouldmapΔLON ontoΔLMN .3. Given:∠HGI ≅ ∠JIG, HG ≅ JI
a. Prove:ΔHGI ≅ ΔJIG
b. Describetherigidmotion(s)thatwouldmapΔJIG ontoΔHGI .
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4. Given: AB PCD, AB ≅ CD a. Prove:ΔABD ≅ ΔCDB
b. Describetherigidmotion(s)thatwouldmapΔCDB ontoΔABD .5. Given: SU andRT bisecteachother
a. Prove:ΔSVR ≅ ΔUVT
b. Describetherigidmotion(s)thatwouldmapΔUVT ontoΔSVR .
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6. Given: JM ≅ KL, JM ⊥ ML, KL ⊥ ML
a. Prove:ΔJML ≅ ΔKLM
b. Describetherigidmotion(s)thatwouldmapΔJML ontoΔKLM .
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Homework1. InordertouseSAStoprovethefollowingtrianglescongruent,drawinthemissing
labels: a b. 2. Given:∠1≅ ∠2, BC ≅ DC
a. Prove:ΔABC ≅ ΔADC
b. Describetherigidmotion(s)thatwouldmapΔADC ontoΔABC .3. Given:KMandJNbisecteachother
a. Prove:ΔJKL ≅ ΔNML
b. Describetherigidmotion(s)thatwouldmapΔNML ontoΔJKL .
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Lesson3:BaseAnglesofIsoscelesTrianglesOpeningExerciseYouwillneedacompassandastraightedgeWearegoingtoshowwhythebaseanglesofanisoscelestrianglearecongruent!Given:IsoscelesΔABC with AB ≅ AC Goal: Toshow∠B ≅ ∠C Step1: Constructtheanglebisectorofthevertex∠ .Step2: ΔABC hasnowbeensplitintotwotriangles. Provethetwotrianglesare≅ .Step3: Identifythecorrespondingsidesandangles.Step4: Whatistrueabout∠B and∠C ?Step5: Whattypesofangleswereformedwhentheanglebisectorintersected BC ?
Whatdoesthismeanabouttheanglebisector?
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What2propertiesdowenowknowaboutisoscelestriangles?1.2.Example1Given:ΔRST isisosceleswith∠R asthevertex,
SY ≅ TZ Prove:ΔRSY ≅ ΔRTZ Onceweprovetrianglesarecongruent,weknowthattheircorrespondingparts(anglesandsides)arecongruent.Wecanabbreviatethisisinaproofbyusingthereasoningof:
CPCTC(CorrespondingPartsofCongruentTrianglesareCongruent).ToProveAnglesorSidesCongruent:
1. Provethetrianglesarecongruent(usingoneoftheabovecriteria)2. Statesthattheangles/sidesarecongruentbecauseofCPCTC.
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Example2Given:ΔJKL isisosceles,KX ≅ LY Prove: JX ≅ JY Example3Given:∠J ≅ ∠M , JA ≅ MB, JK ≅ ML Prove:KR ≅ LR
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Homework1. Given:IsoscelesΔABC with∠A asthevertexangle Disthemidpointof BC
Prove:ΔACD ≅ ΔABD 2. Given: BA ≅ CA , AX istheanglebisectorof∠BAC
Prove:∠ABX ≅ ∠ACX
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Lesson4:CongruenceCriteriaforTriangles–ASAandSSSOpeningExerciseYouwillneedacompassandastraightedge1. Given:ΔABC with∠B ≅ ∠C
Goal: Toprove BA ≅ CA
Step1: Constructtheperpendicularbisectorto BC .
Step2: ΔABC hasnowbeensplitintotwotriangles. ProveBA ≅ CA .
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Thereare5waystotestfortrianglecongruence.Inlesson1wesawthatwecanprovetrianglescongruentusingSAS.Weprovedthisusingrigidmotions.Here’sanotherwaytolookatit:
http://www.mathopenref.com/congruentsas.htmlTodaywearegoingtofocusontwomoretypes:Angle-Side-AngleTriangleCongruenceCriteria(ASA)
• Twopairsofanglesandtheincludedsidearecongruent
Toprovethiswecouldstartwithtwodistincttriangles.WecouldthentranslateandrotateonetobringthecongruentsidestogetherlikewedidintheSASproof(seepicturetotheright).Aswecansee,areflectionoverABwouldresultinthetrianglesbeingmappedontooneanother,producingtwocongruenttriangles.
http://www.mathopenref.com/congruentasa.html
Side-Side-SideTriangleCongruenceCriteria(SSS)
• Allofthecorrespondingsidesarecongruent
Withoutanyinformationabouttheangles,wecannotjustperformareflectionaswedidintheothertwoproofs.Butbydrawinganauxiliaryline,wecanseethattwoisoscelestrianglesareformed,creatingcongruentbaseanglesandtherefore,∠B ≅ ∠B ' .Wecannowperformareflection,producingtwocongruenttriangles.
http://www.mathopenref.com/congruentsss.html
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ExerciseProvethefollowingusinganymethodoftrianglecongruencethatwehavediscussed.Thenidentifytherigidmotion(s)thatwouldmaponetriangleontotheother.1. Given:MisthemidpointofHP ,∠H ≅ ∠P Prove:ΔGHM ≅ ΔRPM Example1ToProveMidpoint/Bisect/Isosceles/Perpendicular/Parallel:
1. Provethetrianglesarecongruent.2. Statethattheangles/sidesarecongruentbecauseofCPCTC.3. Statewhatyouaretryingtoprove.
Given: AB ≅ AC , XB ≅ XC Prove:AX bisects∠BAC
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Example2Given:CircleswithcentersAandBintersectatCandD.Prove:∠CAB ≅ ∠DAB
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HomeworkProvethefollowingusinganymethodoftrianglecongruencethatwehavediscussed.Thenidentifytherigidmotion(s)thatwouldmaponetriangleontotheother.1. Given:∠A ≅ ∠D, AE ≅ DE Prove:ΔAEB ≅ ΔDEC 2. Given: BD ≅ CD ,EisthemidpointofBC Prove:∠AEB ≅ ∠AEC
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Lesson5:CongruenceCriteriaforTriangles–SAAandHLOpeningExerciseWriteaproofforthefollowingquestion.Whenfinished,compareyourproofwithyourpartner’s.Given:DE ≅ DG ,EF ≅ GF Prove:DF istheanglebisectorof∠EDG Wehavenowidentified3differentwaysofprovingtrianglescongruent.Whatarethey?Doesthismeananycombinationof3pairsofcongruentsidesand/orangleswillguaranteecongruence?
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Let’stryanothercombinationofsidesandangles:Side-Angle-AngleTriangleCongruenceCriteria(SAA)
• TwopairsofanglesandasidethatisnotincludedarecongruentToprovethiswecouldstartwithtwodistincttriangles.If∠B ≅ ∠E and∠C ≅ ∠F ,whatmustbetrueabout∠A and∠D ?Why?Therefore,SAAisactuallyanextensionofwhichtrianglecongruencecriterion?
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Let’stakealookattwomoretypesofcriteria:Angle-Angle-Angle(AAA)
• Allthreepairsofanglesarecongruent http://www.mathopenref.com/congruentaaa.htmlDoesAAAguaranteetrianglecongruence?Drawasketchdemonstratingthis.Side-Side-Angle(SSA)
• Twopairsofsidesandanon-includedanglearecongruent http://www.mathopenref.com/congruentssa.htmlDoesSSAguaranteetrianglecongruence?Drawasketchdemonstratingthis.
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ThereisaspecialcaseofSSAthatdoeswork,andthatiswhendealingwithrighttriangles.WecallthisHypotenuse-Legtrianglecongruence.Hypotenuse-LegTriangleCongruenceCriteria(HL)
• Whentworighttriangleshavecongruenthypotenusesandapairofcongruentlegs,thenthetrianglesarecongruent.
Ifweknowtwosidesofarighttriangle,howcouldwefindthethirdside? Therefore,HLisactuallyanextensionofwhichtrianglecongruencecriterion? InordertouseHLtrianglecongruence,youmustfirststatethatthetrianglesare righttriangles!
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ExercisesProvethefollowingusinganymethodoftrianglecongruencethatwehavediscussed.Thenidentifytherigidmotion(s)thatwouldmaponetriangleontotheother.1. Given: AD ⊥ BD, BD ⊥ BC, AB ≅ CD Prove:ΔABD ≅ ΔCDB 2. Given: BC ⊥ CD, AB ⊥ AD, ∠1≅ ∠2 Prove:ΔBCD ≅ ΔBAD
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HomeworkIn1-4,marktheappropriatecongruencemarkingstousethemethodofprovingthatisstated:1. SAS 2. AAS 3. ASA 4. HL5. Given:PA ⊥ AR, PB ⊥ BR, AR ≅ BR Prove:PRbisects∠APB
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Lesson6:TriangleCongruencyProofsOpeningExerciseTriangleproofssummary.Let’sseewhatyouknow!Listthe5waysofprovingtrianglescongruent: 1. 2. 3. 4. 5.WhattwosetsofcriteriaCANNOTbeusedtoprovetrianglescongruent: 1. 2.Inordertoproveapairofcorrespondingsidesoranglesarecongruent,whatmustyoudofirst?Whatistheabbreviationusedtostatethatcorrespondingparts(sidesorangles)ofcongruenttrianglesarecongruent?
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ExercisesProvethefollowingusinganymethodoftrianglecongruencethatwehavediscussed.1. Given: AB ≅ CD BC ≅ DA Prove:ΔADC ≅ ΔCBA 2. Given:NQ ≅ MQ
PQ ⊥ NM Prove:ΔPQN ≅ ΔPQM
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3. Given:∠1≅ ∠2, ∠A ≅ ∠E , CisthemidpointofAE
Prove:BC ≅ DC 4. Given: BD bisects∠ADC
∠A ≅ ∠C Prove:AB ≅ CB
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5. Given: AD bisects BE AB ! DE
Prove: ΔABC ≅ ΔDEC 6. Given:PA ⊥ AR, PB ⊥ BR, AR ≅ BR Prove:PRbisects∠APB
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Homework1. Given: AB !CD, AB ≅ CD
Prove:ΔABD ≅ ΔCDB 2. Given: CD ⊥ AB ,CD bisects AB, AC ≅ BC Prove:ΔACD ≅ ΔBCD
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Lesson7:TriangleCongruencyProofsIIProvethefollowingusinganymethodoftrianglecongruencethatwehavediscussed.1. Given: AB ⊥ BC, BC ⊥ DC DB bisects∠ABC AC bisects∠DCB EB ≅ EC Prove:ΔBEA ≅ ΔCED 2. Given: AB ⊥ BC, DE ⊥ EF, BC P EF, AF ≅ DC
Prove:ΔABC ≅ ΔDEF
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3. Given: AD ⊥ DR, AB ⊥ BR AD ≅ AB Prove:∠ARD ≅ ∠ARB 4. Given: XJ ≅ YK, PX ≅ PY, ∠ZXJ ≅ ∠ZYK Prove: JY ≅ KX
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5. Given:∠1≅ ∠2, ∠3≅ ∠4 Prove: AC ≅ BD
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Homework1. Given:BF ⊥ AC, CE ⊥ AB AE ≅ AF Prove:ΔACE ≅ ΔABF 2. Given: JK ≅ JL, JX ≅ JY Prove:KX ≅ LY
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Lesson8:PropertiesofParallelogramsOpeningExerciseBasedonthediagrampicturedbelow,answerthefollowing:1. Ifthetrianglesarecongruent,statethecongruence.2. Whichtrianglecongruencecriterionguaranteestheyarecongruent?3. SideTGcorrespondswithwhichsideofΔMYJ ?
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Vocabulary
Define DiagramParallelogram
Usingthisdefinitionofparallelogramsandourknowledgeoftrianglecongruence,wecanprovethefollowingpropertiesofparallelograms:
• Oppositesidesarecongruent• Oppositeanglesarecongruent• Diagonalsbisecteachother• Onepairofoppositesidesareparallelandcongruent
Example1Wearegoingtoprovethefollowingsentence: Ifaquadrilateralisaparallelogram,thenitsoppositesidesandanglesareequalin measure.Given: Diagram:Prove:Proof:
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Example2Nowthatwehaveproventhatoppositesidesandanglesofaparallelogramarecongruent,wecanusethatitonourproofs!Wearegoingtoprovethefollowingsentence: Ifaquadrilateralisaparallelogram,thenthediagonalsbisecteachother.Given: Diagram:Prove:Proof:
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Example3Wearegoingtoprovethefollowingsentence: Iftheoppositesidesofaquadrilateralarecongruent,thenthequadrilateralisa parallelogram.Given: Diagram:Prove:Proof:
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HomeworkProvethefollowingsentence: Ifthediagonalsofaquadrilateralbisecteachother,thenthequadrilateralisa parallelogram.Given: Diagram:Prove:Proof:
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Lesson9:PropertiesofParallelogramsIIOpeningExerciseDrawadiagramforeachofthequadrilateralslistedanddrawincongruencemarkingswhereyoubelievetheyexist. Parallelogram Rhombus Rectangle Square
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FamilyofQuadrilateralsManyofthequadrilateralslistedintheOpeningExercisesharesomeofthesameproperties.Wecanlookatthisasafamily:Thequadrilateralsatthebottomhaveallofthepropertiesofthefigureslistedaboveit.Basedonthis,determineifthefollowingaretrueorfalse.Ifitisfalse,explainwhy.
1. Allrectanglesareparallelograms.
2. Allparallelogramsarerectangles.
3. Allsquaresarerectangles. 4. Allrectanglesaresquares.
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Example1Provethefollowingsentence: Ifaparallelogramisarectangle,thenthediagonalsareequalinlength.Given: Diagram:Prove:Proof:
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Example2Provethefollowingsentence: Ifaparallelogramisarhombus,thediagonalsintersectperpendicularly.Given: Diagram:Prove:Proof:
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Homework1. Given:RectangleRSTU,MisthemidpointofRS Prove:ΔUMT isisosceles2. Given:SquareABCS≅ SquareEFGS Prove:ΔASR ≅ ΔESR
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Lesson10:Mid-segmentofaTriangleOpeningExerciseUsingyourknowledgeofthepropertiesofparallelograms,answerthefollowingquestions:1. FindtheperimeterofparallelogramABCD.Justifyyoursolution.2. IfAC=34,AB=26andBD=28,findtheperimeterofΔCED .Justifyyoursolution.
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Vocabulary
Define DiagramMid-segment
Example1YouwillneedacompassandastraightedgeWearegoingtoconstructamid-segment.Steps:
1. ConstructthemidpointsofABandACandlabelthemasXandY,respectively.2. Drawmid-segmentXY.
Compare∠AXY to∠ABC andcompare∠AYX to∠ACB .Withoutusingaprotractor,whatwouldyouguesstherelationshipbetweenthesetwopairsofanglesis?Whataretheimplicationsofthisrelationship?
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PropertiesofMid-segments
• Themid-segmentofatriangleisparalleltothethirdsideofthetriangle.• Themid-segmentofatriangleishalfthelengthofthethirdsideofthetriangle.
ExercisesApplywhatyourknowaboutthepropertiesofmid-segmentstosolvethefollowing:1. a. Findx. b. FindtheperimeterofΔABC 2. Findxandy. 3. Findx.
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Example2Wearenowgoingtoprovethepropertiesofmid-segments.Given:XYisamid-segmentofΔABC
Prove: XY BC and XY = 12BC
Statements Reasons 1.XYisamid-segmentofΔABC 1.Given2.XisthemidpointofAB 2.Amid-segmentjoinsthemidpointsYisthemidpointofAC3. AX ≅ BX and AY ≅ CY 3.4.ExtendXYtopointGsothatYG=XY 4.AuxiliaryLinesDrawGC5.∠AYX ≅ ∠CYG 5.6.ΔAYX ≅ ΔCYG 6.7.∠AXY ≅ ∠CGY , AX ≅ CG 7. 8. BX ≅ CG 8.Substitution9. AB GC 9.10.BXGCisaparallelogram 10.Onepairofopp.sidesare and≅
*11. XY BC 11.Ina ,oppositesidesare
12. XG ≅ BC 12.Ina ,oppositesidesare≅ 13.XG=XY+YG 13.14.XG=XY+XY 14.Substitution15.BC=XY+XY 15.16.BC=2XY 16.Substitution
*17. XY = 12BC 17.
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Homework1. FindtheperimeterofΔEFG . 2. Findandlabelallofthemissing sidesandangles.3. WXisamid-segmentofΔABC ,YZisamid-segmentofΔCWX andBX=AW. a. Whatcanyouconcludeabout∠A and∠B ? Explainwhy. b. WhatistherelationshipinlengthbetweenYZandAB?
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Lesson11:PointsofConcurrencyOpeningExerciseThemidpointsofeachsideofΔRST havebeenmarkedbypointsX,Y,andZ.a. Markthehalvesofeachsidedividedbythemidpointwithacongruencymark. Remembertodistinguishcongruencymarksforeachside.b. Drawmid-segmentsXY,YZ,andXZ.Markeachmid-segmentwiththeappropriate congruencymarkfromthesidesofthetriangle.c. WhatconclusioncanyoudrawaboutthefourtriangleswithinΔRST ?Explainwhy.d. StatetheappropriatecorrespondencesbetweenthefourtriangleswithinΔRST .
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InUnit1wediscussedtwodifferentpointsofconcurrency(when3ormorelinesintersectinasinglepoint).Let’sreviewwhattheyare!Circumcenter
• thepointofconcurrencyofthe3perpendicularbisectorsofatriangle Sketchthelocationofthecircumcenteronthetrianglespicturedbelow:
Incenter
• thepointofconcurrencyofthe3anglebisectorsofatriangle Sketchthelocationoftheincenteronthetrianglespicturedbelow:
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Example1YouwillneedacompassandastraightedgeConstructthemediansforeachsideofthetrianglepicturedbelow.Amedianisasegmentconnectingavertextothemidpointoftheoppositeside.Vocabulary
• Thepointofintersectionfor3mediansiscalledthe___________________.• Thispointisthecenterofgravityofthetriangle.
Wewillusehttp://www.mathopenref.com/trianglecentroid.htmltoexplorewhathappenswhenthetriangleisrightorobtuse.Sketchthelocationofthecentroidonthetrianglesbelow:
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Example2YouwillneedacompassandastraightedgeConstructthealtitudesforeachsideofthetrianglepicturedbelow.Analtitudeisasegmentconnectingavertextotheoppositesideatarightangle.Thiscanalsobeusedtodescribetheheightofthetriangle.Vocabulary
• Thepointofintersectionfor3altitudesiscalledthe_____________________________.Wewillusehttp://www.mathopenref.com/triangleorthocenter.htmltoexplorewhathappenswhenthetriangleisrightorobtuse.Sketchthelocationoftheorthocenteronthetrianglesbelow:
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HomeworkTyisbuildingamodelofahanggliderusingthetemplatebelow.Toplacehissupportsaccurately,Tyneedstolocatethecenterofgravityonhismodel.a. UseyourcompassandstraightedgetolocatethecenterofgravityonTy’smodel.b. ExplainwhatthecenterofgravityrepresentsonTy’smodel.
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Lesson12:PointsofConcurrencyIIOpeningExerciseCompletethetablebelowtosummarizewhatwedidinLesson11.Circumcenterhasbeenfilledinforyou.
PointofConcurrency TypesofSegments Whatthistypeoflineorsegmentdoes
LocatedInsideorOutsideoftheTriangle?
Circumcenter PerpendicularBisectors
Formsarightangleandcutsasideinhalf
Both;dependsonthetypeoftriangle
Incenter
Centroid
Orthocenter
Whichtwopointsofconcurrencyarelocatedontheoutsideofanobtusetriangle?Whatdothesetypeshaveincommon?
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Example1Acentroidsplitsthemediansofatriangleintotwosmallersegments.Thesesegmentsarealwaysina2:1ratio.LabelthelengthsofsegmentsDF,GFandEFasx,yandzrespectively.FindthelengthsofCF,BFandAF.
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Exercises1. Inthefigurepictured,DF=4,BF=16,andGF=10.Findthelengthsof:
a. CF
b. EF
c. AF2. Inthefigureattheright,EF=x+3andBF=5x–9.FindthelengthofEF.3. Inthefigureattheright,DC=15.FindDFandCF.
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Wecannowusemediansandaltitudesintriangleproofs!Here’showitlooks:Given: BD isthemedianofΔABC
Statements Reasons
1. BD isthemedianofΔABC 1. Given 2. 2. 3. 3. Given: BD isthealtitudeofΔABC
Statements Reasons
1. BD isthealtitudeofΔABC 1. Given
2. 2. 3. 3. 4. 4. Example2Given: BD isthemedianofΔABC ,BD ⊥ AC Prove:∠A ≅ ∠C
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Homework1. Inthefigurepictured,DF=3,BF=14,andGF=8.
Findthelengthsof:
a. CF
b. EF
c. AF2. Inthefigureattheright,GF=2x-1andAF=6x–8.FindthelengthofGA.3. Given: BD isthealtitudeofΔABC ,∠ABD ≅ ∠CBD
Prove:ΔABD ≅ ΔCBD