secondary math ii // module 5 geometric figures – 5.3 5 · secondary math ii // module 5...
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SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.3
Mathematics Vision Project
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mathematicsvisionproject.org
5.3
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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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p o3 BI EET Defofbisector4 AT11OFand6THBE RectanglesareParallelograms5 LBEGELECD61 LOBE core Corresponding Angles7 DBOE DEDC ASAE
SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.4
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
5.4 Parallelism Preserved and
Protected
A Solidify Understanding Task
Inaprevioustask,HowDoYouKnowThat,youwereaskedtoexplainhowyouknewthatthisfigure,whichwasformedbyrotatingatriangleaboutthemidpointofoneofitssides,wasaparallelogram.
Youmayhavefounditdifficulttoexplainhowyouknewthatsidesoftheoriginaltriangleanditsrotatedimagewereparalleltoeachotherexcepttosay,“Itjusthastobeso.”Therearealwayssomestatementswehavetoacceptastrueinordertoconvinceourselvesthatotherthingsaretrue.Wetrytokeepthislistofstatementsassmallaspossible,andasintuitivelyobviousaspossible.Forexample,inourworkwithtransformationswehaveagreedthatdistanceandanglemeasuresarepreservedbyrigidmotiontransformationssinceourexperiencewiththesetransformationssuggestthatsliding,flippingandturningfiguresdonotdistorttheimagesinanyway.Likewise,parallelismwithinafigureispreservedbyrigidmotiontransformations:forexample,ifwereflectaparallelogramtheimageisstillaparallelogram—theoppositesidesofthenewquadrilateralarestillparallel.
Mathematicianscallstatementsthatweacceptastruewithoutproofpostulates.Statementsthataresupportedbyjustificationandproofarecalledtheorems.
Knowingthatlinesorlinesegmentsinadiagramareparallelisoftenagoodplacefromwhichtostartachainofreasoning.Almostalldescriptionsofgeometryincludeaparallelpostulateamongthelistofstatementsthatareacceptedastrue.Inthistaskwedevelopsomeparallelpostulatesforrigidmotiontransformations.
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SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.4
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
TranslationsUnderwhatconditionsarethecorrespondinglinesegmentsinanimageanditspre-imageparallelafteratranslation?Thatis,whichwordbestcompletesthisstatement?
Afteratranslation,correspondinglinesegmentsinanimageanditspre-imageare[never,
sometimes,always]parallel.
Givereasonsforyouranswer.Ifyouchoose“sometimes”,beveryclearinyourexplanationabouthowtotellwhenthecorrespondinglinesegmentsbeforeandafterthetranslationareparallelandwhentheyarenot.
RotationsUnderwhatconditionsarethecorrespondinglinesegmentsinanimageanditspre-imageparallelafterarotation?Thatis,whichwordbestcompletesthisstatement?
Afterarotation,correspondinglinesegmentsinanimageanditspre-imageare[never,
sometimes,always]parallel.
Givereasonsforyouranswer.Ifyouchoose“sometimes”,beveryclearinyourexplanationabouthowtotellwhenthecorrespondinglinesegmentsbeforeandaftertherotationareparallelandwhentheyarenot.
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SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.4
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
ReflectionsUnderwhatconditionsarethecorrespondinglinesegmentsinanimageanditspre-imageparallelafterareflection?Thatis,whichwordbestcompletesthisstatement?
Afterareflection,correspondinglinesegmentsinanimageanditspre-imageare[never,
sometimes,always]parallel.
Givereasonsforyouranswer.Ifyouchoose“sometimes”beveryclearinyourexplanationabouthowtotellwhenthecorrespondinglinesegmentsbeforeandafterthereflectionareparallelandwhentheyarenot.
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Dosometimes
SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.4
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
5.4
Needhelp?Visitwww.rsgsupport.org
READY Topic:SpecialQuadrilateralIdentifyeachquadrilateralasatrapezoid,parallelogram,rectangle,rhombus,square,ornoneofthese.ListALLthatapply.
1. 2. 3.
4. 5. 6.
SET Topic:Identifyingparallelsegmentsandlinesproducedfromtransformations7. Verifytheparallelpostulatesbelowbynamingthelinesegmentsinthepre-imageanditsimage
thatarestillparallel.Usecorrectmathematicalnotation.a. Afteratranslation,correspondinglinesegmentsinanimageanditspre-imagearealwaysparallelorliealongthesameline.
b. Afterarotationof180°,correspondinglinesegmentsinapre-imageanditsimageareparallelorlieonthesameline.
READY, SET, GO! Name Period Date
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Rectangleparallelogram Parallelogram trapezoidrhombus
11
Parallelogrampectantpecirallelugram
square Nonepuomws kite
HI HH'T
FHHFFFIHEI.FI
SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.4
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
5.4
Needhelp?Visitwww.rsgsupport.org
c. Afterareflection,linesegmentsinthepre-magethatareparalleltothelineofreflectionwillbeparalleltothecorrespondinglinesegmentsintheimage.
GO Topic:IdentifyingcongruencepatternsintrianglesForeachpairoftriangleswriteacongruencestatementandjustifyyourstatementbyidentifyingthecongruencepatternyouused.Thenjustifythatthetrianglesarecongruentbyconnectingcorrespondingverticesofthepre-imageandimagewithlinesegments.Howshouldthoselinesegmentslook?
8. 9.
10. 11.
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1 Proof2 Classifying Quads
3 Matching terms
4 300 SO