SECONDARY MATH II // MODULE 5

GEOMETRIC FIGURES – 5.3

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5.3 It’s All In Your Head

A Solidify Understanding Task

Intheprevioustaskyouwereaskedtojustifysomeclaimsbywritingparagraphsexplaininghowvariousfigureswereconstructedandhowthoseconstructionsconvincedyouthattheclaimsweretrue.Perhapsyoufounditdifficulttosayeverythingyoufeltyoujustknew.Sometimesweallfinditdifficulttoexplainourideasandtogetthoseideasoutofourheadsandwrittendownorpaper.

Organizingideasandbreakingcomplexrelationshipsdownintosmallerchunkscanmakethetaskofprovingaclaimmoremanageable.Onewaytodothisistouseaflowdiagram.

First,somedefinitions:

• Inatriangle,analtitudeisalinesegmentdrawnfromavertexperpendiculartotheoppositeside(oranextensionoftheoppositeside).

• Inatriangle,amedianisalinesegmentdrawnfromavertextothemidpointoftheoppositeside.

• Inatriangle,ananglebisectorisalinesegmentorraydrawnfromavertexthatcutstheangleinhalf.

• Inatriangle,aperpendicularbisectorofasideisalinedrawnperpendiculartoasideofthetrianglethroughitsmidpoint.

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SECONDARY MATH II // MODULE 5

GEOMETRIC FIGURES – 5.3

Mathematics Vision Project

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Travisusedacompassandstraightedgetoconstructanequilateraltriangle.Hethenfoldedhisdiagramacrossthetwopointsofintersectionofthecirclestoconstructalineofreflection.Travis,Tehani,CarlosandClaritaaretryingtodecidewhattonamethelinesegmentfromCtoD.

Travisthinksthelinesegmenttheyhaveconstructedisalsoamedianoftheequilateraltriangle.Tehanithinksitisananglebisector.ClaritathinksitisanaltitudeandCarlosthinksitisaperpendicularbisectoroftheoppositeside.Thefourfriendsaretryingtoconvinceeachotherthattheyareright.

Onthefollowingpageyouwillfindaflowdiagramofstatementsthatcanbewrittentodescriberelationshipsinthediagram,orconclusionsthatcanbemadebyconnectingmultipleideas.Youwillusetheflowdiagramtoidentifythestatementseachofthestudents—Travis,Tehani,CarlosandClarita—mightusetomaketheircase.Togetreadytousetheflowdiagram,answerthefollowingquestionsaboutwhateachstudentneedstoknowaboutthelineofreflectiontosupporttheirclaim.

1. Tosupporthisclaimthatthelineofreflectionisamedianoftheequilateraltriangle,Traviswillneedtoshowthat:

2. Tosupportherclaimthatthelineofreflectionisananglebisectoroftheequilateraltriangle,Tehaniwillneedtoshowthat:

3. Tosupportherclaimthatthelineofreflectionisanaltitudeoftheequilateraltriangle,Claritawillneedtoshowthat:

4. Tosupporthisclaimthatthelineofreflectionisaperpendicularbisectorofasideoftheequilateraltriangle,Carloswillneedtoshowthat:

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AT BFL ACD EL BTD

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CDIAB TAE DD

SECONDARY MATH II // MODULE 5

GEOMETRIC FIGURES – 5.3

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

Hereisaflowdiagramofstatementsthatcanbewrittentodescriberelationshipsinthediagram,orconclusionsthatcanbemadebyconnectingmultipleideas.

7. Usefourdifferentcolorstoidentifythestatementseachofthestudents—Travis,Tehani,ClaritaandCarlosmightusetomaketheircase.

Given:ΔABCisequilateral AND

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AB ≅ BC ≅ AC

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∠CDA and

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∠CDB arerightangles

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CD⊥ AB

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CD ≅ CD

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CDisananglebisector

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CDisaperpendicularbisector

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SECONDARY MATH II // MODULE 5

GEOMETRIC FIGURES – 5.3

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

8. Matcheachofthearrowsandbracesintheflowdiagramwithoneofthefollowingreasonsthatjustifieswhyyoucanmaketheconnectionbetweenthestatement(orstatements)previouslyacceptedastrueandtheconclusionthatfollows:

1. Definitionofreflection2. Definitionoftranslation3. Definitionofrotation4. Definitionofanequilateraltriangle5. Definitionofperpendicular6. Definitionofmidpoint7. Definitionofaltitude8. Definitionofmedian9. Definitionofanglebisector10. Definitionofperpendicularbisector11. Equilateraltrianglescanbefoldedontothemselvesaboutalineofreflection12. Equilateraltrianglescanberotated60°ontothemselves13. SSStrianglecongruencecriteria14. SAStrianglecongruencecriteria15. ASAtrianglecongruencecriteria16. Correspondingpartsofcongruenttrianglesarecongruent17. ReflexiveProperty

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SECONDARY MATH II // MODULE 5

GEOMETRIC FIGURES – 5.3

Mathematics Vision Project

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Travisandhisfriendshaveseentheirteacherwritetwo-columnproofsinwhichthereasonsjustifyingastatementarewrittennexttothestatementbeingmade.Travisdecidestoturnhisargumentintoatwo-columnproof,asfollows.

Statements Reasons

ΔABCisequilateral Given

isalineofreflection Equilateraltrianglescanbefoldedontothemselvesaboutalineofreflection

Disthemidpointof Definitionofreflection

isamedian Definitionofmedian

9. WriteeachofClarita’s,Tehani’s,andCarlos’argumentsintwo-columnproofformat.

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SECONDARY MATH II // MODULE 5

GEOMETRIC FIGURES – 5.3

Mathematics Vision Project

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5.3

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READY Topic:Congruencestatementsandcorrespondingparts

Rememberthatwhenyouwriteacongruencestatementsuchas∆!"# ≅ ∆!"#,thecorrespondingpartsofthetwotrianglesmustbethepartsthatarecongruent.

Forinstance,∠A ≅∠F, AB ≅ FG,∠B ≅∠G, BC ≅GH .Also,recallthatthecongruencepatternsfortriangles,ASA.SAS,andSSS,arewhatwecanusetojustifytrianglecongruence.

Thesegmentsandanglesineachproblembelowarecorrespondingpartsof2congruenttriangles.Makeasketchofthetwotriangles.Thenwriteacongruencestatementforeachpairoftrianglesrepresented.Statethecongruencepatternthatjustifiesyourstatement.

Congruencestatement Congruencepattern

1. ML ≅ ZJ, LR ≅ JB,∠L ≅ ∠J a. b.

2. WB ≅QR, BP ≅ RS,WP ≅QS a. b.

3. CY ≅ RP, EY ≅ BP,∠Y ≅ ∠P a. b.

4. BC ≅ JK , BA ≅ JM ,∠B ≅ ∠J a. b.

5. DF ≅ XZ, FY ≅ ZW ,∠F ≅ ∠Z a. b.

6. WX ≅ AB, XZ ≅ BC,WZ ≅ AC a. b.

READY, SET, GO! Name Period Date

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Bf Dmp eDEB SAS

SECONDARY MATH II // MODULE 5

GEOMETRIC FIGURES – 5.3

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SET Topic:Specialtrianglesegmentsandproof.Recallthefollowingdefinitions:

Inatriangle:• analtitudeisalinesegmentdrawnfromavertexperpendicularto

theoppositeside(oranextensionoftheoppositeside).

• amedianisalinesegmentdrawnfromavertextothemidpointoftheoppositeside.

• ananglebisectorisalinesegmentorraydrawnfromavertexthatcutstheangleinhalf.

• aperpendicularbisectorofasideisalinedrawnperpendiculartoasideofthetrianglethroughitsmidpoint.

Besuretousethecorrectnotationforasegmentinthefollowingproblems.

7. Nameasegmentin∆!"# thatisanaltitude.

8. Nameasegmentin∆!"#thatisananglebisector.

9. Nameasegmentin∆!"#thatisNOTanaltitude.

10. Createaperpendicularbisectorbymarkingtwosegmentscongruentin∆!"#.Namethesegmentthatisnowtheperpendicularbisector.

Use∆!"#inproblems11–13.

11. ConstructthealtitudefromvertexDto .12. ConstructthemedianfromDto .13. Constructtheperpendicularbisectorof .

EF

EFEF

F

E

D

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SECONDARY MATH II // MODULE 5

GEOMETRIC FIGURES – 5.3

Mathematics Vision Project

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Tehanihasbeenstudyingthefigurebelow.SheknowsthatquadrilateralADEGisarectangleandthatEDbisects BC .Sheiswonderingifwiththatinformationshecanprove∆!"# ≅ ∆!"#.Shestartstoorganizeherthinkingbywritingwhatsheknowsandthereasonssheknowsit.

IknowED bisects BC becauseIwasgiventhatinformationIknowthatBE ≅ EC bydefinitionofbisect.IknowthatGE mustbeparallelto AD becausetheoppositesidesinarectangleareparallel.Iknowthat GA ED becausetheyareoppositesidesinarectangle.IknowthatAD iscontainedin AC so AC isalsoparalleltoGE .IknowthatGA iscontainedin BA soGA isalsoparallelto BAIknowthat BC

hasthesameslopeeverywherebecauseitisaline.

Iknowtheanglethat BE makeswithGE mustbethesameastheanglethatEC makeswith ACsincethose2segmentsareparallel.So∠!"# ≅ ∠!"#.IthinkIcanusethatsameargumentfor∠!"# ≅ ∠!"#.IknowthatInowhaveanangle,aside,andananglecongruenttoacorrespondingangle,side,andangle.So∆!"# ≅ ∆!"#byASA.

14. UseTehani’s“Iknow”statementsandherreasonstowriteatwo-columnproofthatproves∆!"# ≅ ∆!"#.Beginyourproofwiththe“givens”andwhatyouaretryingtoprove.

Given:quadrilateralADEGisarectangle,ED bisectsProve:∆!"# ≅ ∆!"#

STATEMENTS REASONS1. quadrilateralADEGisarectangle given2. ED bisects given

AC

AC

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