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Unit6–Quadrilaterals
Day Classwork Day HomeworkFriday11/18
PropertiesofaParallelogram
1 HW6.1
Monday11/21
ProvingaParallelogram
2 HW6.2
Tuesday11/22
Rectangle
3 HW6.3
Wednesday11/23
Rhombus&SquareUnit6Quiz1
4 HW6.4
11/24‐11/27
ThanksgivingBreak
Monday11/28
Trapezoid&IsoscelesTrapezoid
5 HW6.5
Tuesday11/29
KitesUnit6Quiz2
6 HW6.6
Wednesday11/30
CoordinateProofFormulas 7 HW6.7
Thursday12/1
CoordinateProofs 8 HW6.8
Friday12/2
SymmetryinQuadrilaterals
9 HW6.9
Monday12/5
ReviewUnit6Quiz3
10 ReviewSheet
Tuesday12/6
Review
11 ReviewSheet
Wednesday12/7
Unit6Test 12
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PARALLELOGRAMSAparallelogramisaquadrilateralwithbothpairsofoppositesidesparallel.
InparallelogramABCD, and bydefinition.
PropertiesofParallelogramsTheorems Example Figure
Oppositesidesofaparallelogramare
congruent.
Given:JKLMisaparallelogram
Prove: ,JK LM JM LK
Oppositeanglesofaparallelogramare
congruent.
Given:JKLMisaparallelogramProve: ,J L K M
J K
M L
J K
M L
J K
M L
J K
M L
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ConsecutiveAnglesinaparallelogramaresupplementary.
Examples1. InparallelogramABCD, and .Find .
2. InparallelogramABCD, and .Findx.
3. InparallelogramABCD,AB=7x–3andCD=2x+22.Findthevalueofx.
Diagonalsofa
parallelogrambisecteachother
Given:JKLMisaparallelogramProve: JL andKM bisecteachother
J K
M L
J K
M L
J K
M L
P
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Examples
1. IfQRSTisaparallelogram,findthevalueofx,y,andz.
2. InparallelogramABCD,diagonals and intersectatE.
IfAE=x+4andAC=5x–10,findthevalueofx.
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PROVINGPARALLELOGRAMS
Ifaquadrilateralhaseachpairofoppositesidesparallel,itisaparallelogrambydefinition.Thisisnottheonlytest,however,thatcanbeusedtodetermineifaquadrilateralisa
parallelogram.
ConditionsforParallelogramsTheorem Example Figure
Ifbothpairsofoppositesidesarecongruent,thenthequadrilateralisa
parallelogram.
Given: ,AB CD BC DA Prove:ABCDisaparallelogram
Ifbothpairsofoppositeangles
arecongruent,thenthequadrilateralisaparallelogram.
Given: ,A C B D Prove:ABCDisaparallelogram
A B
CD
A B
CD
A B
CD
A B
CD
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Ifdiagonalsbisecteach
other,thenthequadrilateralisaparallelogram.
Given: AB andC D bisecteachotherProve:ABCDisaparallelogram
Ifonepairofoppositesidesiscongruentandparallel,thenthequadrilateralisa
parallelogram.
Given: AB CD , AB CD Prove:ABCDisaparallelogram
A B
CD
A B
CD
A B
CD
A B
CD
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ExamplesDeterminewhethereachquadrilateralisaparallelogram.Justifyyouranswer.1. 2. 3.Findxandysothateachofthefollowingquadrilateralsareparallelograms.4. FK=3x–1,KG=4y+3,JK=6y–2,andKH=2x+35. 6.
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RECTANGLESBydefinition,arectangleisaparallelogramwithfourrightangles.
1. Allfouranglesarerightangles 4.Consecutiveanglesaresupplementary2. Oppositesidesare and 5.Diagonalsbisecteachother
3. Oppositeanglesare 6.Diagonalsofarectangleare
DiagonalsofaRectangle
Theorem Example Figure
Arectangleisaparallelogramwithcongruentdiagonals
Given:ABCDisarectangleProve: AC BD Examples
1. InrectangleJKLM,JL=2x+15andKM=4x–5.FindMP.
2. QuadrilateralJKLMisarectangle.If =2x+4and =7x+5,findx.
A B
CD
A B
CD
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3. Given:ABCDisarectangle.
Prove:
ProvingParallelogramsareRectanglesAbbreviation Example Figure
Ifaparallelogramhasonerightangle,thenithasfourright
angles.
Ifdiagonalsofaparallelogram
arecongruent,thentheparallelogramisarectangle.
Given: AC BD ,ABCDisaparallelogramProve:ABCDisarectangle
A B
CD
A B
CD
A B
CD
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RHOMBIANDSQUARES
Arhombusisaparallelogramwithallfoursidescongruent.Arhombushasallthepropertiesofaparallelogram.
DiagonalsofaRhombus
Theorem Example Figure
Ifaparallelogramisarhombus,thenitsdiagonals
areperpendicular
Given:ABCDisarhombusProve: AC BD
Ifaparallelogramisarhombus,theneach
diagonalbisectsapairofoppositeangles.
Given:ABCDisarhombusProve: A C bisects BAD and BCD
BD bisects ABC and ADC
A B
CD
A B
CD
A B
CD
A B
CD
P
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ExamplesThediagonalsofrhombusFGHIintersectatK.Usethegiveninformationtofindeachmeasureorvalue.
a. If =82,find .
b. IfKH=x+5,KG=x–2,andFG=17.FindKH.
Asquareisaparallelogramwithfourcongruentsidesandfourrightangles.Recallthataparallelogramwithfourrightanglesisarectangle,andaparallelogramwithfourcongruentsidesisarhombus.Therefore,aparallelogramthatisbotharectangleandarhombusisalso
asquare.
ConditionsforRhombiandSquaresTheorem Example Figure
Ifthediagonalsofaparallelogramare
perpendicular,thentheparallelogramisarhombus.
Ifonepairofconsecutivesidesofaparallelogramare
congruent,thentheparallelogramisarhombus.
Ifaquadrilateralisbotharectangleandarhombus,thenitisasquare.
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TRAPEZOIDSAtrapezoidisaquadrilateralwithatleastonepairofparallelsides.Theparallelsidesarecalledbases.Thenonparallelsidesarecalledlegs.Thebaseanglesareformedbythebaseandoneofthelegs.Byadefinition,anisoscelestrapezoidisatrapezoidwithatleastonepair
ofoppositesidescongruent.
IsoscelesTrapezoidsTheorem Example Figure
Ifatrapezoidisisosceles,theneachpairofbaseanglesarecongruent
Ifatrapezoidhasonepairofcongruentbaseangles,thenitisanisosceles
trapezoid.
Atrapezoidisisoscelesifandonlyifitsdiagonalsare
congruent.
Given:ABCDisanisoscelestrapezoidwith AD BC Prove: ,A B C D
A B
CD
A B
CD
A B
CD
A B
CD
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Given:ABCDisatrapezoidand C D Prove:ABCDisanisoscelestrapezoidGiven:ABCDisatrapezoidand AC BD Prove:ABCDisanisoscelestrapezoid
A B
CD
A B
CD
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Examples1. Thespeakershownisanisoscelestrapezoid.If =85,FK=8inches,andJG=19
inches,findeachmeasure.
c.
d. KH
2. Tosavespaceatasquaretable,cafeteriatraysoftenincorcporatetrapezoidsintotheir
design.IfWXYZisanisoscelestrapezoidand =45,WV=15cm,andVY=10cm,findeachmeasurebelow.a. b.
c.XZ d.XVThemidsegmentofatrapezoidisthesegmentthatconnectsthemidpointsofthelegsofthe
trapezoid.Thetheorembelowrelatesthemidsegmentandthebasesofatrapezoid.
TrapezoidMidsegmentTheoremTheorem Example Figure
Themidsegmentofa
trapezoidisparalleltoeachbaseanditsmeasureisonehalfthesumofthelengthsof
thebases.
Examples1. Inthefigure,PQRSisatrapezoid. isthemedian.IfSR=2x–3,PQ=2x+11,andTU=14,whatisthelengthof ?
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ADDITIONALPRACTICEPROOFS
1. Given: , .
Prove: isarhombus
2. Given:Rectangle , isthemidpointof .
Prove: isisosceles.
3. Given: isaparallelogram, bisects ,
bisects .
Prove: isaparallelogram.
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4. Given: isarectangle, , .
Prove: isaparallelogram.
5. Given:Parallelogram , .
Prove: .
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KITES
Akiteisaquadrilateralwithatleasttwopairsofconsecutivecongruentsides.Unlikeaparallelogram,theoppositesidesofakitearenotcongruentorparallel.
Theorem Example Figure
Ifaquadrilateralisakite,thenitsdiagonalsare
perpendicular.
Ifaquadrilateralisakite,thenatleastonepairofoppositeanglesare
congruent.
Examples
a. IfFGHJisakite,find .
b. IfWXYZisakite,findZY.
c. If and ,find .
d. IfBT=5andTC=8,findCD.
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SLOPESOFLINES
Slopecanbeinterpretedasrateofchange,describinghowaquantityychangesinrelationshiptoquantityx.Theslopeofalinecanalsobeusedtoidentifythecoordinatesof
anypointontheline.
Examples
1. Determinewhether and areparallel,perpendicular,orneitherforA(1,1),B(‐1,‐5),C(3,2),andD(6,1).
2. GivenA(1,1),B(2,4),C(4,1),andD(3,k)a) Findtheslopeof AB
.
b) ExpresstheslopeofCD
intermsofk.
c) If AB CD
,findthevalueofk.
ParallelandPerpendicularLines Description Example
SlopesofParallelLines
SlopesofPerpendicular
Lines
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DistanceThedistancebetweentwopointsisthelengthofthesegmentwiththosepointsasits
endpoints.
Examples
1. FindthedistancebetweenC(1,1)andD(3,‐3).CheckusingthePythagoreanTheorem.
2. GiventhepointsA(6,7)andB(14,‐1).Findthelengthof AB .
3. Whatisthelengthofthediameterofacirclewhosecenterisat(6,0)andpassesthrough(2,‐3)?
4. AtrianglehasverticesD(2,3),E(5,5),andF(4,0).Determineifthetriangleisscalene,isosceles,orequilateral.
DISTANCEFORMULA(CoordinatePlane)
WORDS SYMBOLS PICTURE
IfPhascoordinates(x1,y1)andQhascoordinates(x2,y2),then
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MidpointThemidpointofasegmentisthepointhalfwaybetweentheendpointsofthesegment.
Examples
1. FindthecoordinatesofM,themidpointof ,forS(‐6,3)andT(2,1).
2. Findthemidpointwhengiventheendpoints(‐1,‐4)and(3,‐2).
3. FindthecoordinatesofJifM(‐1,2)isthemidpointofandLhascoordinates(3,‐5).
4. RS isthediameterofthecircleshownintheaccompanyingdiagram.Whatarethecoordinatesofthecenterofthiscircle?
MidpointFormula(CoordinatePlane)
WORDS SYMBOLS PICTURE
If hasendpointsatP(x1,y1)andQ(x2,y2)inthecoordinateplane,then
themidpointMof is
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COORDINATEPROOF
ParallelLines()–Iftwolineshaveequalslopes,thenthelinesareparallel.
PerpendicularLines( )–Iftwonon‐verticallineshaveslopesthatarenegativereciprocalsofoneanother,thenthelinesareperpendicular.Examples:1. GiventhepointsA(6,9)andB(14,‐1).
a. Findtheslopeof AB .
b. Findtheslopeofthelineperpendicularto AB
c. Findtheslopeofthelineparallelto AB
2. Considerthelinesegments AB andCD ,withA(‐3,‐2),B(2,1),C(‐7,‐1),andD(‐2,2).
a. Provethat AB CD .
b. Determineif BC AB .
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3. IfX(5,0),Y(3,4),andZ(‐1,2),prove XY YZ .4. TheverticesoftriangleWINareW(2,1),I(4,7)andN(8,3).Usingcoordinate
geometry,showthat∆WINisanisoscelestriangleandstatethereasonsforyourconclusion.
5. TriangleNAQhascoordinatesN(2,3),A(6,0)andQ(12,8).Usingcoordinategeometry,
showthat∆NAQisarighttriangleandstatethereasonsforyourconclusion
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CLASSIFYINGQUADRILATERALSUSINGCOORDINATEGEOMETRY
1. DeterminethecoordinatesoftheintersectionofthediagonalsofparallelogramFGHI
withverticesF(‐2,4),G(3,5),H(2,‐3),andJ(‐3,‐4).ProveFGHIisaparallelogram.2. GraphquadrilateralKLMNwithverticesK(2,3),L(8,4),M(7,‐2),andN(1,‐3).Prove
thequadrilateralisaparallelogram.JustifyyouranswerusingtheSlopeFormulaandDistanceformula.
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3. QuadrilateralPQRShasverticesP(‐5,3),Q(1,‐1),R(‐1,‐4),andS(‐7,0).ProvePQRSisa
rectangle.4. ProvethatparallelogramJKLMwithverticesJ(‐7,‐2),K(0,4),L(9,2),and
M(2,‐4)isarhombus.Isitarectangleand/orsquarealso?
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5. A(‐3,4),B(2,5),C(3,3),D(‐1,0).ProveABCDisatrapezoid.Isthetrapezoidisosceles?6. F(‐2,4),G(3,5),H(2,‐3),J(‐3,‐4).ProvethatquadrilateralFGHJisaparallelogram.
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SYMMETRYINQUADRILATERALS
RECALL:Afigurehassymmetryifthereexistsarigidmotion–reflection,translation,rotation,orglide‐reflection–thatmapsthefigureontoitself.
Afigureintheplanehaslinesymmetryifthefigurecanbemappedontoitselfbyareflectioninaline,calleda
lineofsymmetry.
Anontrivialrotationalsymmetryofafigureisarotationoftheplanethatmapsthefigurebacktoitselfsuchthat
therotationisgreaterthan0˚butlessthan360˚.
1. Determinewhethereachfigurehaslinesymmetryand/orrotationalsymmetry.Ifso,drawalllinesofsymmetryand/orgivetheangleofrotationalsymmetry.
a. Rectangle b.IsoscelesTrapezoid
c. Parallelogram d.RegularHexagon
2. SupposeABCDisaquadrilateralforwhichthereisexactlyonerotation,throughananglelargerthan0degreesandlessthan360degrees,whichmapsittoitself.Further,noreflectionsmapABCDtoitself.WhatshapeisABCD?
3. Drawanexampleofatrapezoidthatdoesnothavelinesymmetry.
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4. JenniferdrawstherectangleABCDbelow.a. FindallrotationsandreflectionsthatcarryrectangleABCDontoitself.
b. Lisadrawsadifferentrectangleandshefindsalargernumberofsymmetries(thanJennifer)forherrectangle.WhatcanyouconcludeaboutLisa'srectangle?Explain.
5. ThereisexactlyonereflectionandnorotationthatsendstheconvexquadrilateralABCDontoitself.Whatshape(s)couldquadrilateralABCDbe?Explain.
6. Drawanexampleofaparallelogramthathasexactlytwolinesofsymmetry.Drawthelinesofsymmetryandgivethemostspecificnamefortheparallelogramyoudrew.