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 Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 5.3 Need help? Visit www.rsgsupport.org Tehani has been studying the figure below. She knows that quadrilateral ADEG is a rectangle and that ED bisects BC . She is wondering if with that information she can prove ∆!"# ≅ ∆!"# . She starts to organize her thinking by writing what she knows and the reasons she knows it. I know ED bisects BC because I was given that information I know that BE ≅ EC by definition of bisect. I know that GE must be parallel to AD because the opposite sides in a rectangle are parallel. I know that GA ED because they are opposite sides in a rectangle. I know that AD is contained in AC so AC is also parallel to GE . I know that GA is contained in BA so GA is also parallel to BA I know that BC has the same slope everywhere because it is a line. I know the angle that BE makes with GE must be the same as the angle that EC makes with AC since those 2 segments are parallel. So ∠!"# ≅ ∠!"#. I think I can use that same argument for ∠!"# ≅ ∠!"# . I know that I now have an angle, a side, and an angle congruent to a corresponding angle, side, and angle. So ∆!"# ≅ ∆!"# by ASA. 14. Use Tehani’s “I know” statements and her reasons to write a two-column proof that proves ∆!"# ≅ ∆!"# . Begin your proof with the “givens” and what you are trying to prove. Given: quadrilateral ADEG is a rectangle, ED bisects Prove: ∆!"# ≅ ∆!"# STATEMENTS REASONS 1. quadrilateral ADEG is a rectangle given 2. ED bisects given AC AC Page 22 p r f p o 3 BI EET Def of bisector 4 AT 11 OFand 6TH BE Rectangles are Parallelograms 5 LBEGELECD 61 LOBE core Corresponding Angles 7 D BOE D EDC ASA E
Transcript of SECONDARY MATH II // MODULE 5 GEOMETRIC FIGURES – 5.3 5 · SECONDARY MATH II // MODULE 5...
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
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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
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Parallelogrampectantpecirallelugram
square Nonepuomws kite
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FHHFFFIHEI.FI
SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.4
Mathematics Vision Project
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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
