secondary math ii // module 5 geometric figures – 5.3 5 · secondary math ii // module 5...

SECONDARY MATH II // MODULE 5

GEOMETRIC FIGURES – 5.3

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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

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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

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5.4

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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

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SECONDARY MATH II // MODULE 5

GEOMETRIC FIGURES – 5.4

Mathematics Vision Project

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mathematicsvisionproject.org

5.4

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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