GeometryHonors-Chapter3–ParallelandPerpendicularLinesSection1–ParallelLinesandTransversals

• Icanpreciselydefinelinesegments,rays,parallellines,perpendicularlines,andskewlinesanddescribetheircharacteristics.

• Icanidentifyandnameanglepairsformedbyparallellinesandtransversals(corresponding,alternateinterior,alternateexterior,andconsecutiveinterior).

• ParallelLines–coplanarlinesthatdonotintersectortouch.• SkewLines–arelinesthatdonointersectandarenon-coplanar(meanstheyarenotparallelorintersecting).• ParallelPlanes–planesthatdonotintersectortouch.

1. Identifyeachofthefollowingusingtheboxbelow:

(a) allsegmentsparallelto𝐵𝐶 (b)asegmentskewto𝐸𝐻 (c)aplaneparalleltoplaneABG

• Transversal–alinethatcutsorintersects2ormorecoplanarlinesat2differentpoints(ex:linet)

Homework–Page174–176(13-19ODD,21-43,50,51)

Section2–AnglesandParallelLines• Icanproveandapplytheoremsabouttheanglesformedbyparallellinesandatransversal

(corresponding,alternateinterior,alternateexterior,andconsecutiveinterior).

• Correspondinganglespostulate–If2parallellinesarecutbyatransversal,thencorrespondingangles

arecongruent.(ex:∠1≅∠3,∠8≅∠6;∠2≅∠4,∠7≅∠5)• AlternateInterioranglestheorem–If2parallellinesarecutbyatransversal,thenalternateinterior

anglesarecongruent.(ex:∠2≅∠6,∠3≅∠7)• AlternateExterioranglestheorem–If2parallellinesarecutbyatransversal,thenalternateexterior

anglesarecongruent.(ex:∠1≅∠5,∠4≅∠8)• ConsecutiveInterioranglestheorem–If2parallellinesarecutbyatransversal,thenconsecutive

interioranglesaresupplementary(2anglesthataddto180.)(ex:m∠2+m∠3=180,m∠7+m∠6=180)1.Inthefigure,m∠11=51.Findthemeasureofeachangle.Tellwhichpostulatesortheoremsyouused.

(a)m∠15 (b)m∠162.ThefollowingdiagramrepresentsfloortilesinMichelle’shouse.Ifm∠2=125,findm∠3.

3.Usethefigurebelowtofindtheindicatedvariable.Explainyourreasoning.

(a)Ifm∠5=2x–10andm∠7=x+15,findx. (b)Findy,ifm∠6=4(y–25)andm∠8=4y.

• PerpendicularTransversalTheorem–inaplane,ifalineisperpendicularto1of2parallellines,thenit

isperpendiculartotheother.

4.

5.

6.

Homework–Page181–183(11–30,38,39,42,43,45)

Section5–ProvingLinesParallel• Icanproveandapplytheoremsabouttheanglesformedbyparallellinesandatransversal

(corresponding,alternateinterior,alternateexterior,andconsecutiveinterior).

• ConverseofCorrespondinganglespostulate–if2linesarecutbyatransversalsothatcorresponding

anglesarecongruent,thenthelinesareparallel.(ex:If∠1≅∠3,thena∥b; If∠8≅∠6,thena∥b;If∠2≅∠4,thena∥b; If∠7≅∠5,thena∥b)

• AlternateExterioranglesconverse–if2linesarecutbyatransversalsothatalternateexterioranglesarecongruent,thenthelinesareparallel (ex:If∠1≅∠5,thena∥b OR∠8≅∠4,thena∥b)

• AlternateInterioranglesconverse–if2linesarecutbyatransversalsothatalternateinterioranglesarecongruent,thentheliensareparallel (ex:If∠2≅ ∠6,thena∥b OR ∠7≅∠3,thena∥b)

• ConsecutiveInterioranglesconverse-if2linesarecutbyatransversalsothatconsecutiveinterioranglesaresupplementary,thenthelinesareparallel. (ex:m∠2+m∠3=180,thena∥b ORm∠7+m∠6=180,thena∥b)

• PerpendicularTransversalconverse–if2linesareperpendiculartothesameline,thenthe2linesareparallel.

1.Giventhefollowinginformation,isitpossibletoprovethatanyofthelinesshownareparallel?Ifso,statethepostulateortheoremthatjustifiesyouranswer.

(a)∠1≅∠3 (b)m∠1=103andm∠4=100

#1Guidedpractice:Giventhefollowinginformation,isitpossibletoprovethatanyofthelinesshownareparallel?Ifso,statethepostulateortheoremthatjustifiesyouranswer:

(a)∠2≅∠8

(b)∠3≅∠11 (c)∠12≅∠14(d)∠1≅∠15(e)m∠8+m∠13=180(f)∠8≅∠62.(a)Findm∠𝑍YNsothat𝑃𝑄 ∥ 𝑀𝑁.Showyourwork.

(b)Findysothate∥f.Showyourwork.

3.Inordertomoveinastraightlinewithmaximumefficiency,rower’soarsshouldbeparallel.Refertothephotoattheright.Isitpossibletoprovethatanyoftheoarsareparallel?Ifso,explainhow.Ifnot,explainwhynot.

FIREWORKS A fireworks display is being readied for a celebration. The designers want to have four fireworks shoot out along parallel trajectories. They decide to place two launchers on a dock and the other two on the roof of a building. To pull off this display, what should the measure of angle 1 be?

Homework–Page209-211(8–21,23,33–35,37,42)ALL

Section3–SlopesofLines• Icanfindslopesoflines.

• Icanuseslopetoprovelinesareparallelorperpendicular.

1.Findtheslopeofeach:(a)(6,-2)and(-3,-5) (b)(4,2)and(4,-3)(c)(8,-3)and(-6,-2) (d)(-3,3)and(4,3)

Find the slope of each line.

Slope: also known as a rate of change, the change in y over the change in x.

2.In2000,theannualsalesforonemanufacturerofcampingequipmentwas$48.9million.In2005,thetotalsaleswere$85.9million.Ifsalesincreaseatthesamerate,whatwillbethetotalsalesin2015?3.After Take Two began renting DVDs at their video store, business soared. Between 2005 and 2010, profits increased at an average rate of $9000 per year. Total profits in 2010 were $45,000. If profits continue to increase at the same rate, what will the total profit be in 2014?

• ParallelLines–Iflinesareparallel,theyhavethesameorequalslopes.(ex:ifa∥b,slopea=slopeb)If

ma=½thenmb=½• PerpendicularLines–iflinesareperpendicular,thentheyhaveslopeswithaproductof-1.(ex:ifa⊥b,

slopea*slopeb=-1)Ifma=½,themb=-2

4.

5.Graph the line that satisfies each condition

Homework–Page191–193(15-33ODD,34-39ALL,41,48,50,52,53,55)

Section4–EquationsofLines• Icanfindtheequationofalineparallelorperpendiculartoagivenlinethatpassesthroughagivenpoint.

• Slope–InterceptForm:y=mx+b,where‘m’istheslopeand‘b’isthey-intercept(wheretheline

crossesthey-axis) (ex:m=3andb=-2 so y=3x–2)

• Point-SlopeForm:y–y1=m(x–x1),where‘m’istheslopeand(x1,y1)isapointontheline. (ex:m=½and(3,5)isapointontheline so y–5=½(x–3))1.Writeanequationinslope-interceptformgivenaslopeanday-intercept.Thengraphtheline.(a)m=6andy-intercept=-3(b)m=½andb=8

2.Writeanequationinpoint-slopeformgivenaslopeandapointontheline.Thengraphtheline.(a)m=-3/5andcontains(-10,8)(b)m=4andcontains(-3,-6)

3.Writeanequationofalinethrougheachpairofpointsinslope-interceptform:(a)(4,9)and(-2,0)(b)(-3,-7)and((-1,3)

Horizontal&VerticalLineEquationsTheequationofahorizontallineisy=bwherebisthey-intercept.Allhorizontallineshavepointsthatusethesamey-coordinateHorizontallineshaveaslopeofzero.Theequationofaverticallineisx=awhereaisthex-intercept.Allverticallineshavepointsthatusethesamex-coordinate.Allverticallineshaveaslopethatisundefined.Whatistheequationofalinethathasaslopeof0andpassesthrough(-3,8)?Thengraphtheline.

Whatistheequationofalinethathasanundefinedslopeandpassesthrough(-1,4)?Thengraphtheline.

KeyThingstoRemember:*m=0isperpendiculartom=undefined

*m=0isahorizontallineandtheequationisy=b(itnevercrossesthex-axis)*m=undefinedisaverticallineandtheequationisx=a(itnevercrossesthey-axis)

Example:

-Writeanequationinslope-interceptformof(3,2)and(1,2)

-Writeandequationinslope-interceptformof(-2,6)and(-2,-1)

4. Writeanequationofthelinethrough(5,-2)and(0,-2)inslope-interceptform.

5. Writeanequationinslope-interceptformforalineperpendiculartotheliney=1/5x+2through(2,0)6.Writeanequationinslope-interceptformforalineparalleltotheliney=-3/4x+3andcontaining(-3,6)7.Anapartmentcomplexcharges$525permonthplusa$750annualmaintenancefee.(a)Writeanequationtorepresentthetotalfirstyear’scost‘A’for‘r’monthsofrent.(b)Comparethisrentalcosttoacomplexwithnoannualmaintenancefeebut$600permonthforrent.Ifapersonexpectstostayinanapartmentforoneyear,whichcomplexoffersthebetterrate?

Homework–Page200-202(13-29ODD,32,38,40,41,46,49,51,52,56,58)Section6–PerpendicularsandDistance

• Icanfindthedistancebetweenapointandalineandbetweenparallellines.

• Distancebetweenapointandaline–thedistancebetweenalineandapointnotonthelineisthelengthofthesegmentperpendiculartothelinefromthepoint.

• PerpendicularPostulate–ifgivenalineandapointnotontheline,thenthereexistsexactly1linethroughthepointthatisperpendiculartotheline.Example:

1. ConstructandnamethesegmentthatrepresentsthedistancefromQto𝑃𝑅(a)distancefromQto𝑃𝑅 (b)distancefromYtoTS

(c)distancefromCto𝐴𝐵

2.(a)Linescontainspointsat(0,0)and(-5,5).FindthedistancebetweenlinesandpointV(1,5).

2.(b)Linelcontainspointsat(1,2)and(5,4).ConstructalineperpendiculartolthroughP(1,7).ThenfindthedistancefromPtol.

Distance Between Parallel Lines The distance between parallel lines is the length of a segment that has an endpoint on each line and is perpendicular to them. Parallel lines are everywhere equidistant, which means that all such perpendicular segments have the same length.

Find the distance between each pair of parallel lines with the given equations.

Find the distance between each pair of parallel lines with the given equation.

Hint:Ifyouhavetwolinesthatareparallelandtheperpendicularsegmentdoesnotlandonapointthatyoucanread,youwillhavetowritetheequationofthelinefortheperpendicularsegmentandsolvetofindthepointthattheyhaveincommon.

3.(a)Findthedistancebetweentheparallellineswithequationsy=2x+3andy=2x–1,respectively.

3.(b)Findthedistancebetweenparallellineswithequationsx+3y=6andx+3y=-15,respectively.

Homework–Page218–221(9–12ALL,15–25ODD)