1

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