5.3 it’s all in your head //congleton.weebly.com/uploads/8/8/7/7/88774672/...secondary math ii //...
TRANSCRIPT
SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.3
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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.
CC
BY
tec
_est
rom
berg
http
s://f
lic.k
r/p/
hWT
88P
Page 15
so
is 19
SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.3
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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:
Page 16
AT BFL ACD EL BTD
EDL FB
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
�
CE isalineofreflection
�
AB ≅ BC ≅ AC
�
∠CDA and
�
∠CDB arerightangles
�
CD⊥ AB
Disthemidpointof
�
AB
�
AD ≅ DB
�
CD ≅ CD
�
∠ACD ≅ ∠BCD ΔACD≅ΔBCD
therefore,
�
CDisanaltitude
therefore,
�
CDisamedian
therefore,
�
CDisananglebisector
therefore,
�
CDisaperpendicularbisector
Page 17
4 b I I
617
f
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
Page 18
i
SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.3
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Travisandhisfriendshaveseentheirteacherwritetwo-columnproofsinwhichthereasonsjustifyingastatementarewrittennexttothestatementbeingmade.Travisdecidestoturnhisargumentintoatwo-columnproof,asfollows.
Statements Reasons
ΔABCisequilateral Given
isalineofreflection Equilateraltrianglescanbefoldedontothemselvesaboutalineofreflection
Disthemidpointof Definitionofreflection
isamedian Definitionofmedian
9. WriteeachofClarita’s,Tehani’s,andCarlos’argumentsintwo-columnproofformat.
�
CE
�
AB
�
CD
Page 19
SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.3
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
5.3
Needhelp?Visitwww.rsgsupport.org
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
Page 20
Bf Dmp eDEB SAS
SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.3
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
5.3
Needhelp?Visitwww.rsgsupport.org
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
Page 21
SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.3
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
5.3
Needhelp?Visitwww.rsgsupport.org
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
Page 22