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November2016 ©LewDouglasandHenriPicciotto SymmetryDefinitionsandProperties- TrianglesandQuadrilaterals Thisdocumentisareferenceforteachersandcurriculumdevelopers.Itisbasedon achoicetorethinkthegeometrycurriculumonatransformationalfoundation.Itisa sequeltoour“TriangleCongruenceandSimilarity:ACommon-Core-Compatible Approach”(availableonhttp://www.mathedpage.org/transformations/).Asyou willsee,thisapproachimpliessomechangesinthehierarchyofquadrilaterals. WerecommendthatyoureadTransformationProofBasicsfirst.Itcontainsthe definitions,assumptions,andlemmas(simple,helpingtheorems)onwhichthese proofsarebased. 1. IsoscelesTriangle:Atrianglewithonelineofsymmetry. Theetymologyof“isosceles”,ofcourse,is“equallegs”.Intheschemewe propose,thisisnolongerthedefinition:itmustbeproved.(SeePropertyc below.) Properties: a. Theimageofavertexinalineofsymmetryisalsoavertex. Proof:Avertexisthecommonendpointoftwosides.Because collinearityispreserved,sidesmustmapontosides.Sotheimageofa vertexmustalsolieontwosides.Apointontwosidesisavertex,soit mustalsobeavertex. b. Onevertexliesonthelineofsymmetryandtheothertwoareeach other'sreflections. Proof:Becausethereisanoddnumberofvertices,oneofthemmust lieonthelineofsymmetry. c. Anisoscelestrianglehastwoequalsidesandtwoequalangles. Proof:Reflectionpreservessidelengths andanglemeasure.IfvertexAisonthe lineofsymmetry,thenAB=ACand ∠B=∠C. www.MathEducationPage.org p.1 November2016 ©LewDouglasandHenriPicciotto d. Theperpendicularbisectorofthethirdsideofanisoscelestriangle bisectsanangleofthetriangle,sothelineofsymmetryisanaltitude,a median,andaperpendicularbisector. Proof:Bydefinitionofreflection,theline ofsymmetrylistheperpendicular bisectorofBC.ItalsomustpassthroughA. Sincereflectionpreservesangles, ∠DAB=∠DAC.Therefore,ADisan altitude,amedian,andaperpendicular bisector. 2. EquilateralTriangle:Atrianglewithtwolinesof symmetry. Other(equivalent)definitionsarepossible. Wepreferthisone,asitiseconomical,and facilitatestheproofofproperties. Notethatonceagain,theetymologydoesnot correspondtothedefinition:thatthesidesare equalmustbeproved.(Seepropertybbelow.) Properties: a. Anequilateraltrianglehas3-foldrotationalsymmetry. Proof:LetmandnbethesymmetrylinesthroughAandC respectively.Thecompositionofthereflectionsinmandnmapsthe triangleontoitselfandisarotationaroundtheirintersectionpointD. Callthisrotationr.rmapsAontoB,BontoC,andContoA.Repeating thisrotationthreetimesgivestheidentitytransformation,sothe trianglehas3-foldrotationalsymmetryaroundtheintersectionpoint ofitstwolinesofsymmetry. b. Allsidesofanequilateraltriangleareequalandeachangleis60˚. Proof:Rotationpreservessidelengthsandanglemeasure.Sincethe sumoftheanglesinatriangleis180˚,eachangleis60˚. c. Anequilateraltrianglehasthreeconcurrentlinesofsymmetry. Proof:rmapsAtoBandDtoitself,so m',theimageofmunderr,passes throughBandD.Sincem perpendicularlybisectsBC,m'must perpendicularlybisectCAbecause rotationpreservessegmentlengthand anglemeasure.Thereforem'isathird lineofsymmetryofΔABC. www.MathEducationPage.org p.2 November2016 ©LewDouglasandHenriPicciotto d. Eachlineofsymmetryofanequilateraltriangleisanaltitude,a median,andaperpendicularbisector. Proof:Thetriangleisisoscelesinthreedifferentways. 3. Parallelogram:Aquadrilateralwith2-foldrotationalsymmetry. Thisistheonlyspecialquadrilateralwhosedefinitiondoesnotinvolveline symmetry. Properties: a. Theimageofavertexunderthesymmetryrotationisanopposite vertex. Proof:Letrbethe2-foldrotation.risnottheidentityandrfollowed byr(rr)isa360°rotation,i.e.theidentity.Aswithtriangles,the imageofavertexunderrmustbeavertex.Itsimageunderrrmustbe itself.Iftheimagewereaconsecutivevertex,thentheimageunderrr wouldbethenextconsecutivevertex(i.e.theoppositevertex),notthe original.Therefore,theimageistheoppositevertex. (Thisargumentissubtle,andtheresultisobviousenoughthatwe recommendnotincludingitindiscussionswithstudents.Theywillbe willingtoacceptthisresultwithoutaproof,andwouldmostlikely findtheproofmoreconfusingthanilluminating.Thesameistrueof property(d)below.) b. Thecenterofthe2-foldrotationisthecommonmidpointofthe diagonals. Proof:Adiagonalmustrotateintoitselfbecauseitsendpointsswitch. Soadiagonalmustcontainthecenterofrotation.Thedistancefrom thecentertoonediagonalendpointmustequalthedistancetothe otherbecauserotationpreservesdistance.Therefore,thecentermust bethecommonmidpointofbothdiagonals. c. Theoppositesidesofaparallelogramareparallel. Proof:Theimageofalineunderahalf-turnaroundapointnotonthe lineisaparallelline. d. Theimageofasideunderrisanoppositeside.Theimageofanangle underrisanoppositeangle. Proof:Theimagecan'tbeaconsecutivesidebecausethenitwouldn't beparalleltothepre-image.Theimageofananglecan'tbea consecutiveanglebecausethenoneimagesidewouldn'tbeparallelto itspre-image. 4. Kite:Aquadrilateralwithonelineofsymmetrythroughoppositevertices.(It wouldbepossibletoomit“opposite”fromthedefinition,andinsteadprove thatifthelineofsymmetrypassesthroughvertices,theymustbeopposite www.MathEducationPage.org p.3 November2016 ©LewDouglasandHenriPicciotto vertices.Butformoststudentsthissortofsubtletycouldbepedagogically counterproductive.Ontheotherhand,itcouldbea“bonus”exercisefora strongerstudent.) Properties: a. Akitehastwodisjointpairsofconsecutiveequalsidesandonepairof equaloppositeangles. Proof:Theimageandpre-imageofsidesandanglesunderreflection inthelineofsymmetryhaveequallength(sides)andequalmeasure (angles). b. Thelineofsymmetryofakitebisectsapairofoppositeangles. Proof:Reflectionpreservesanglemeasure. c. Thediagonalofakitethatliesonthelineofsymmetry perpendicularlybisectstheotherdiagonal. Proof:Thesymmetrylineperpendicularlybisectsthesegmentjoining pre-imageandimageoftheverticesnotontheline. 5. IsoscelesTrapezoid:Aquadrilateralwithalineofsymmetrythoughinterior pointsofoppositesides.Thesesidesarecalledbases.Theothertwosidesare calledlegs. Properties: a. Twoverticesofanisoscelestrapezoidareononesideofthe symmetrylineandtwoareontheother. Proof:Sincereflectionmapsverticestovertices,thefourvertices mustbeevenlysplitonbothsidesofthesymmetryline. b. Thesymmetrylineofanisoscelestrapezoidistheperpendicular bisectorofthetwooppositesidesthroughwhichitpasses. Proof:Oneendpointofeachofthesesidesmustreflectintotheother. Areflectionlineperpendicularlybisectsthesegmentjoiningpreimageandimagepointsifthesepointsarenotonthereflectionline. c. Thebasesofanisoscelestrapezoidareparallel. Proof:Theyarebothperpendiculartothesymmetryline.Twodistinct linesperpendiculartothesamelineareparallel. d. Thelegsofanisoscelestrapezoidareequal. Proof:Reflectionpreservessegmentlength. e. Twoconsecutiveanglesofanisoscelestriangleonthesamebaseare equal. Proof:Reflectionpreservesanglemeasure. www.MathEducationPage.org p.4 November2016 ©LewDouglasandHenriPicciotto f. Thediagonalsofanisoscelestrapezoidareequal. Proof:Onediagonalreflectstotheother.Reflectionpreserves segmentlength. g. Theintersectionpointoftheequaldiagonalsofanisoscelestrapezoid liesonthesymmetryline. Proof:Thepointwhereonediagonalintersectsthesymmetryline mustbeinvariantunderreflectioninthesymmetrylinebecauseitlies onit.Thereforeitalsoliesontheotherdiagonal. h. Theintersectionpointofthediagonalsofanisoscelestrapezoid divideseachdiagonalintoequalsubsections. Proof:Thesubsectionsofonediagonaldeterminedbytheintersection pointreflectontothesubsectionsoftheother.Thesesubsectionsare equalbecausereflectionpreservessegmentlength. 6. Rhombus:Aquadrilateralwithtwolinesofsymmetrypassingthrough oppositevertices.(Soarhombusisakiteintwodifferentways.) Properties: a. Arhombushasallsidesequalandtwopairsofequaloppositeangles. Proof:Akitehastwodisjointpairsofconsecutiveequalsidesandone pairofequaloppositeangles.Sincearhombusisakiteintwo differentways(i.e.bothdiagonalsarelinesofsymmetry),theresult follows. b. Eachdiagonalofarhombusbisectsitsangles. Proof:Eachlineofsymmetrybisectsapairofoppositeangles (propertyofkites). c. Thediagonalsofarhombusperpendicularlybisecteachother. Proof:Thediagonalofakitethatliesonthelineofsymmetry perpendicularlybisectstheotherdiagonal.Forarhombus,each diagonalhasthisproperty. d. Arhombusisaspecialparallelogram. Proof:Sincearhombushastwoperpendicularlinesofsymmetry,the compositionofreflectioninthoseyieldsa180˚rotationaroundtheir pointofintersectionthatmapstherhombustoitself.(The compositionoftworeflectionsisarotationaroundtheirpointof intersectionthroughtwicetheanglebetweenthereflectionlines.) e. Theoppositesidesofarhombusareparallel. Proof:Sincearhombusisaparallelogram,theoppositesidesare parallel. www.MathEducationPage.org p.5 November2016 ©LewDouglasandHenriPicciotto 7. Rectangle:Aquadrilateralwithtwolinesofsymmetrypassingthrough interiorpointsoftheoppositesides.(Soarectangleisanisoscelestrapezoid intwodifferentways.) Properties: a. Thesymmetrylinesofarectangleperpendicularlybisecttheopposite sides. Proof:arectangleisanisoscelestrapezoidintwodifferentways. b. Arectangleisequiangular. Proof:Twoconsecutiveanglesofanisoscelestrapezoidthatsharea baseareequal.Bothpairsofoppositesidesarebasesbecauseofthe twodifferentways,soanytwoconsecutiveanglesshareabase. c. Allanglesofarectanglearerightangles. Proof:Thesumoftheinterioranglesofanyquadrilateralis360˚and 360÷4=90. d. Thesymmetrylinesofarectangleareperpendicular. Proof:Thelinesdividetherectangleintofourquadrilaterals.Eachhas threerightangles:oneisanangleoftherectangleandtheothertwo areformedbyasideandasymmetryline,whichareperpendicular. Sincethesumoftheanglesofaquadrilateralis360˚,thefourthangle attheintersectionofthesymmetrylinesmustalsobearightangle. e. Arectanglehas2-foldrotationalsymmetry,soitisaspecial parallelogram. Proof:Reflectingarectangleinonelineofsymmetryfollowedbythe othermapstherectangleontoitselfandisequivalenttoa180˚ rotationbecausethesymmetrylinesmeetatrightangles.Thereforea rectanglehas2-foldrotationalsymmetryaroundtheintersectionof thesymmetrylines. f. Theoppositesidesofarectangleareparallelandequal. Proof:Thesearepropertiesofaparallelogram.Arectangleisaspecial parallelogram. g. Thediagonalsofarectangleareequal. Proof:Thisisapropertyofanisoscelestrapezoid.Arectangleisa specialisoscelestrapezoid. h. Thediagonalsofarectanglebisecteachother. Proof:Thisisapropertyofaparallelogram.Arectangleisaspecial parallelogram. www.MathEducationPage.org p.6 November2016 ©LewDouglasandHenriPicciotto i. Thediagonalsofarectangleandthelinesofsymmetryareall concurrent. Proof:Theintersectionpointoftheequaldiagonalsofanisosceles trapezoidliesonthesymmetryline.Forarectangle,theintersection pointliesonbothsymmetrylines,soitistheirintersection. 8. Square:Aquadrilateralwithfourlinesofsymmetry:twodiagonalsandtwo linespassingthroughinteriorpointsofoppositesides. Properties: a. Asquareisaspecialrectangle,rhombus,kite,andisoscelestrapezoid, soitinheritsallthepropertiesofthesequadrilaterals. Proof:Truebydefinitionofasquare. b. Ifasquareandallfoursymmetrylinesaredrawn,alltheacuteangles are45˚. Proof:Thediagonalsbisecttheinteriorrightanglesbecauseasquare isarhombus.Alleightrighttrianglesformedhavearightanglewhere thesymmetrylinesintersectthesidesanda45˚anglewherethey intersectthevertices.Sincethesumoftheanglesofatriangleis180˚, theremaininganglesatthecentermustallbe45˚. c. Asymmetrylinethroughsidesandasymmetrylinethroughvertices forma45˚angle. Proof:Animmediateconsequenceoftheresultjustabove. d. Asquarehas4-foldrotationalsymmetry. Proof:Reflectingasquareinalineofsymmetrythroughthesides followedbyalineofsymmetrythroughtheverticesmapsthesquare ontoitself.Itisequivalenttoa90˚rotationbecausethesesymmetry linesmeetata45˚angle.Therefore,asquarehas4-foldrotational symmetryarounditscenter(theintersectionpointofthelinesof symmetry). www.MathEducationPage.org p.7