Unit 6 Notes - Quadrilaterals - Revize€¦ · Unit 6 – Quadrilaterals Day Classwork Day Homework...

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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


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.


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.

Unit 6 Notes - Quadrilaterals - Revize€¦ · Unit 6 – Quadrilaterals Day Classwork Day Homework...

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