Download Symmetry Defs and Properties

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