parallel lines lines that intersect are in a common plane. euclid's fifth postulate 18 lines that...

2

GeometryParallelLines

20161115

www.njctl.org

http://www.njctl.org

3

TableofContentsClickonthetopictogotothatsection

Lines:Intersecting,Parallel&Skew

ConstructingParallelLines

Lines&Transversals

ParallelLines&Proofs

PropertiesofParallelLines

PARCCSampleQuestions

4

ConstructionsVideosTableofContentsClickonthetopictogotothatvideo

ParallelLinesCorrespondingAngles

ParallelLinesAlternateInteriorAngles

ParallelLinesAlternateExteriorAngles

ParallelLinesusingMenuOptions

5

Throughoutthisunit,theStandardsforMathematicalPracticeareused.

MP1:Makingsenseofproblems&persevereinsolvingthem.MP2:Reasonabstractly&quantitatively.MP3:Constructviableargumentsandcritiquethereasoningofothers.MP4:Modelwithmathematics.MP5:Useappropriatetoolsstrategically.MP6:Attendtoprecision.MP7:Lookfor&makeuseofstructure.MP8:Lookfor&expressregularityinrepeatedreasoning.

Additionalquestionsareincludedontheslidesusingthe"MathPractice"Pulltabs(e.g.ablankoneisshowntotherightonthisslide)withareferencetothestandardsused.

Ifquestionsalreadyexistonaslide,thenthespecificMPsthatthequestionsaddressarelistedinthePulltab.

MathPractic

e

6

Lines:Intersecting,Parallel&Skew

ReturntoTableofContents

7

Euclid'sFifthPostulate

Euclid'sFifthPostulateisperhapshismostfamous.

It'sbotheredmathematiciansforthousandsofyears.

FifthPostulate:That,ifastraightlinefallingontwostraightlinesmaketheinterioranglesonthesamesidelessthantworightangles,thetwostraightlines,ifproducedindefinitely,meetonthatsideonwhicharetheangleslessthanthetwo

rightangles.

8


ThisseemedsonaturalthattheGreekgeometersthoughttheyshouldbeabletoproveit,andwouldn'tneedittobeapostulate.

Theyresistedusingitforyears.

However,theyfoundthattheyneededit.

Andtheycouldn'tproveit.

Theyjusthadtopostulateit.

9

Itsaysthattherearetwopossiblecasesifonelinecrossestwoothers.


1 2

3 4

10

Thepairsofanglesonbothsides,(either1&3or2&4)eachaddupto180,tworightangles,andthetworedlinesnevermeet.

Likethis....


1 2

3 4

11

Orlikethis.

1 2

3 4


Or,...

12

Theyadduptolessthan180ononeside(angles2&4),andmorethan180ontheother(angles1&3),inwhichcasethelinesmeetonthesidewiththesmallerangles.

Likethis...

1 2

3 4


13

Orlikethis.


1 2

3 4

14

Theycouldn'tprovethisfromtheotheraxiomsandpostulates.But,withoutittherewerealotofimportantpiecesofgeometrythey

couldn'tprove.

SotheygaveinandmadeitthefinalpostulateofEuclideanGeometry.Forthenextthousandsofyears,mathematiciansfeltthesameway.Theykepttryingtoshowwhythispostulatewasnot

needed.

Noonesucceeded.


15

In1866,BernhardRiemanntooktheotherperspective.

ForhisdoctoraldissertationhedesignedageometryinwhichEuclid'sFifthPostulatewasnottrue,ratherthanassumingitwas.

ThisledtononEuclideangeometry.Whereparallellinesalwaysmeet,ratherthannevermeet.


16

Buthalfacenturylater,nonEuclideangeometry,basedonrejectingthefifthpostulate,becamethemathematicalbasisofEinstein'sGeneralRelativity.

Itcreatestheideaofcurvedspacetime.Thisisnowtheacceptedtheoryfortheshapeofouruniverse.


17

Linesthatareinthesameplaneandnevermeetarecalledparallel.

Linesthatintersectarecallednonparallelorintersecting.Alllinesthatintersectareinacommonplane.


18

Linesthatareindifferentplanesandnevermeetarecalledskew.


m

n

P

Q

Linesm&ninthefigureareskew.

19

B

a

TheParallelPostulate

OnewayofrestatingEuclid'sFifthPostulateistosaythatparallellinesnevermeet.

AnextensionofitistheParallelPostulate:givenalineandapoint,notontheline,thereisone,andonlyone,linethatcanbedrawnthroughthepointwhichisparalleltotheline.

Canyouestimatewhere

theparallellinewouldbe?

MathPractic

e

QuestiononthisslideaddressesMP2

20

B

a

CanyouimagineanyotherlinewhichcouldbedrawnthroughPointBandstillbeparalleltolinea?

TheParallelPostulate

MathPractic

e

QuestiononthisslideaddressesMP2

21

Parallellinesaretwolinesinaplanethatnevermeet.

WewouldsaythatlinesDEandFGareparallel.

Or,symbolically:

Parallel,IntersectingandSkew

DE FG

D E

F G

MathPractic

e

22

Linescannotbeassumedtobeparallelunlessitisindicatedthattheyare.Justlookingliketheyareparallelisnotsufficient.

Therearetwowaysofindicatingthatlinesareparallel.

Thefirstwayisasshownonthepriorslide:

IndicatingLinesareParallel

D E

F G

DE FG

23


m

k

Theotherwaytoindicatelinesareparallelistolabelthemwitharrows,asshownbelow.

Thelineswhichsharethearrow(showninredtomakeitmorevisiblehere)areparallel.

Iftwodifferentpairsoflinesareparallel,theoneswiththematchingnumberofarrowsareparallel,asshownonthenextslide.

MathPractic

e

24


m

k

a

b

Thisindicatesthatlineskandmareparalleltoeachother.

And,linesaandbareparalleltoeachother.

Butlineskandmarenotparalleltoaandb.

25

Iftwodifferentlinesinthesameplanearenotparalleltheyareintersecting,andtheyintersectatonepoint.

Wealsoknowthatfouranglesareformed.


D

E

F G

26

Fromthesefourangles,therearefourpairsoflinearanglesthatareformedorlinearpairs.

Linearpairsareadjacentanglesformedbyintersectinglinestheanglesaresupplementary.

1&3areonelinearpair


D

E

F G1 2

3 4

Listtheotherlinearpairs.

Ans

wer

27

Iftheadjacentanglesformedbyintersectinglinesarecongruent,thelinesareperpendicular.

PerpendicularLines

D

E

F G

DE FGSymbolically,thisisstatedas

MathPractic

e

28

Iftwolinesintersect,thentheydefineaplane,soarecoplanar.

SkewLines

m

n Q

PLinesm&ninthefigureareskew.

Twolinesthatdonotintersectcaneitherbeparalleliftheyareinthesameplaneorskewiftheyareindifferentplanes.

MathPractic

e

29

Usingthefollowingdiagram,namealinewhichisskewwithLineHG:alinethatdoesnotlieinacommonplane.

A B

CD

E F

GH

SkewLines

Ans

wer

ThisexampleaddressesMP2

30

1 Arelinesaandbskew?

Yes No

a

b

G Ans

wer Yes,linesaandbareskew.Linesaandbare

noncoplanaranddonotintersect.

31

2 HowmanylinescanbedrawnthroughCandparalleltoLineAB?

B

AC

Ans

wer

32

A B

CD

E F

GH

3 NamealllinesparalleltoEF.

A ABB BCC DCD HDE HG

Ans

wer

33

4 NamelinesskewtoEF.

A BCB DCC HDD ABE GC

A B

CD

E F

GH

Ans

wer

34

5 Twointersectinglinesarealwayscoplanar.

True False

Ans

wer

35

6 Twoskewlinesarecoplanar.

True False

Ans

wer

36

7 Completethisstatementwiththebestappropriateword:

Twoskewlinesare__________parallel.

A alwaysB neverC sometimes

Ans

wer

37

Lines&Transversals


38

123 4

56

7 8

k

m

n

Transversals

(ThisisthenameofthelinethatEuclidusedtointersecttwolinesinhisfifthpostulate.)

Intheimage,transversal,Linen,isshownintersectingLinekandLinem.

ATransversalisalinethatintersectstwoormorecoplanarlines.

LinekandLinemmayormaynotbeparallel.

39

InteriorAnglesarethe4anglesthatliebetweenthetwolines.

Whenatransversalintersectstwolines,eightanglesareformed.Theseanglesaregivenspecialnames.

3 4

56

k

m

n

AnglesFormedbyaTransversal

40

ExteriorAnglesarethe4anglesthatlieoutsidethetwolines.

Whenatransversalintersectstwolines,eightanglesareformed.Theseanglesaregivenspecialnames.

12

7 8

k

m

n

AnglesFormedbyaTransversal

41

8 Namealloftheinteriorangles.

A 1

B 2

C 3

D 4

E 5

F 6

G 7

H 8

123 4

567 8

k

m

n

Ans

wer

42

9 Namealloftheexteriorangles.

A 1

B 2

C 3

D 4

E 5

F 6

G 7

H 8

123 4

567 8

k

m

n

Ans

wer

43

CorrespondingAnglesarepairsofanglesthatlieinthesamepositionrelativetothetransversal,asshownabove.

CorrespondingAngles

123 4

567 8

k

n

m

Therearefourpairsofcorrespondinganglesformedwhenatransversalintersectstwolines.

44

10 Whichanglecorrespondswith1?

123 4

567 8

k

m

n

Ans

wer

45


123 4

567 8

k

m

n

Ans

wer

46


123 4

567 8

k

m

n

Ans

wer

47


123 4

567 8

k

m

n

Ans

wer

48

Therearetwopairsformedbythetransversaltheyareshownaboveinredandblue.

AlternateInteriorAngles

3 4

56

k

m

n

AlternateInteriorAnglesareinterioranglesthatlieonoppositesidesofthetransversal.

MathPractic

e

MP6Emphasizebreakingapartthewordsineachvocabularytermtounderstandthemeaning.

Alternatemeans"opposite"Interiormeans"inside"

SoAlternateInteriorAnglesareonoppositesidesofthetransversalandinsideoftheother2lines.

49

AlternateExteriorAngles

12

7 8

k

m

n


AlternateExteriorAnglesareexterioranglesthatlieonoppositesidesofthetransversal.

MathPractic

e


Alternatemeans"opposite"Exteriormeans"outside"

SoAlternateExteriorAnglesareonoppositesidesofthetransversalandoutsideoftheother2lines.

50

14 Whichisthealternateinterioranglethatispairedwith3?

A 1B 2C 3D 4

E 5F 6G 7H 8

123 4

567 8

k

m

n

Ans

wer

51

15 Whichisthealternateexterioranglethatispairedwith7?

123 4

567 8

k

m

nA 1B 2C 3D 4E 5F 6G 7H 8

Ans

wer

52

123 4

567 8

k

m

n

16 Whichisthealternateexterioranglethatispairedwith2?

A 1B 2C 3D 4E 5F 6G 7H 8

Ans

wer

53

123 4

567 8

k

m

n

17 Whichisthealternateinterioranglethatispairedwith6?

A 1B 2C 3D 4E 5F 6G 7H 8

Ans

wer

D

54

SameSideInteriorAngles


3 4

56

k

m

n

SameSideInteriorAnglesareinterioranglesthatlieonthesamesideofthetransversal.

MathPractic

e


Samesidemeans"onthesameside",Interiormeans"inside"SoSameSideInteriorAngles

areonsamesideofthetransversalandinsideoftheother2lines.Note:a.k.a.

ConsecutiveInteriorAngles

55

SameSideExteriorAngles


12

7 8

k

m

n

SameSideExteriorAnglesareexterioranglesthatlieonthesamesideofthetransversal.

MathPractic

e

56

123 4

567 8

k

m

n

18 Whichisthesamesideinterioranglethatispairedwith6?

A 1B 2C 3D 4E 6F 7G 8

Ans

wer

C

57

123 4

567 8

k

m

n

19 Whichisthesamesideexterioranglethatispairedwith7?

A 1B 2C 3D 4E 6F 7G 8

Ans

wer

B

58

123 4

567 8

k

m

n

a.1and2

b.1and3

c.1and5

d.3and6

e.3and5

f.2and8

Slideeachwordintotheappropriatesquaretoclassifyeachpairofangles.

ClassifyingAngles

AlternateExteriorSameSideInteriorVertical

CorrespondingSameSideExterior AlternateInteriorLinearPair

Ans

wer

59

1 23 4

5 67 8

kj

t

20 3and6are...

A CorrespondingAnglesB AlternateExteriorAnglesC SameSideExteriorAnglesD VerticalAnglesE Noneofthese

Ans

wer

B

60

1 23 4

5 67 8

kj

t

21 1and6are____.


Ans

wer

C

61

1 23 4

5 67 8

kj

t

22 2and7are____.

A CorrespondingAnglesB AlternateInteriorAnglesC SameSideInteriorAnglesD VerticalAnglesE Noneofthese An

swer

B

62

1 23 4

5 67 8

kj

t

23 4and8are____.


Ans

wer

A

63

1 23 4

5 67 8

kj

t

24 1and7are____.A CorrespondingAnglesB AlternateExteriorAnglesC SameSideExteriorAnglesD VerticalAnglesE Noneofthese

Ans

wer E

Theseangleshavenorelationshipw/oneanother

64

1 23 4

5 67 8

kj

t

25 5and8are____.


Ans

wer

D

65

1 23 4

5 67 8

kj

t

26 2and5are____.

A CorrespondingAnglesB AlternateInteriorAnglesC SameSideInteriorAnglesD VerticalAnglesE Noneofthese

Ans

wer

C

66

ParallelLines&Proofs


Lab:StartingaBusinessWorksheet

Lab:StartingaBusinessTeacherSlides

MathPractic

e

ThislabaddressesMP1,MP3,MP4,MP6&MP7

http://www.njctl.org/courses/math/geometry/parallel-lines/starting-a-business-lab-2/http://www.njctl.org/courses/math/geometry/parallel-lines/starting-a-business-lab-slides-2/

67

Inadditiontothepostulatesandtheoremsusedsofar,therearethreeessentialpropertiesofcongruenceuponwhichwe

willrelyasweproceed.

Therearealsofourpropertiesofequality,threeofwhicharecloselyrelatedtomatchingpropertiesofcongruence.

PropertiesofCongruenceandEquality

68

TheyallrepresentthesortofcommonsensethatEuclidwouldhavedescribedasaCommonUnderstanding,andwhichwe

wouldnowcallanAxiom.

Thecongruencepropertiesaretrueforallcongruentthings:linesegments,anglesandfigures.

Theequalitypropertiesaretrueforallmeasuresofthingsincludinglengthsoflinesandmeasuresofangles.

PropertiesofCongruenceandEquality

69

Athingisalwayscongruenttoitself.

Whilethisisobvious,itwillbeusedinprovingtheoremsasareason.

Forinstance,whenalinesegmentservesasasideintwodifferenttriangles,youcanstatethatthesidesofthosetrianglesarecongruentwiththereason:

ReflexivePropertyofCongruence

ReflexivePropertyofCongruence

A B

CD

Inthediagram,ACAC

70

Themeasuresofanglesorlengthsofsidescanbetakentobeequaltothemselves,eveniftheyarepartsof

differentfigures,

withthereason:

ReflexivePropertyofEquality

ReflexivePropertyofEquality

A B DC

TheLineSegmentAdditionPostulatetellusthat

AC=AB+BCandBD=CD+BC

TheReflexivePropertyofEqualityindicatesthatthelengthBCisequaltoitselfinbothequations

71

SymmetricPropertyofCongruence

Ifonethingiscongruenttoanother,thesecondthingisalsocongruenttothefirst.

Again,thisisobviousbutallowsyoutoreversetheorderofthestatementsaboutcongruentpropertieswiththereason:

SymmetricPropertyofCongruence

Forexample:

ABCiscongruentto DEFthat DEFiscongruentto ABC,

72

SymmetricPropertyofEquality

Ifonethingisequaltoanother,thesecondthingisalsoequaltothefirst.

Again,thisisobviousbutallowsyoutoreversetheorderofthestatementsaboutequalpropertieswiththereason:

SymmetricPropertyofEquality

Forexample:

Ifm ABC=m DEF,thenm DEF=m ABC,

73

Iftwothingsarecongruenttoathirdthing,thentheyarealsocongruenttoeachother.

So,ifABCiscongruenttoDEFandLMNisalsocongruenttoDEF,thenwecansaythatABCiscongruenttoLMNduetothe

Withthereason:

TransitivePropertyofCongruence

TransitivePropertyofCongruence

74

Iftwothingsareequaltoathirdthing,thentheyarealsoequaltoeachother.

IfmA=mBandmC=mB,thenmA=mC

Thisisidenticaltothetransitivepropertyofcongruenceexceptitdealswiththemeasureofthingsratherthanthethings.

TransitivePropertyofEquality

TransitivePropertyofEquality

75

Ifonethingisequaltoanother,thenonecanbesubstitutedforanother.

Thisisacommonstepinaproofwhereonethingisprovenequaltoanotherandreplacesthatotherinanexpressionusingthereason:

SubstitutionPropertyofEquality

Forinstanceifx+y=12,andx=2y

Wecansubstitute2yforxtoget

2y+y=12

andusethedivisionpropertytogety=4

SubstitutionPropertyofEquality

76

123 4

567 8

k

m

n

CorrespondingAnglesTheorem

AccordingtotheCorrespondingAngleswhichoftheaboveanglesarecongruent?

Ifparallellinesarecutbyatransversal,thenthecorrespondinganglesarecongruent.

Ans

wer

AccordingtoCorrespondingAnglesPostulatethefollowinganglesare

congruent:

15263748

77

Wecouldpickanypairofcorrespondingangles:2&63&71&5or4&8.

Together,let'sprovethat2&6arecongruent.

CorrespondingAnglesProof

Tokeeptheargumentclear,let'sjustproveonepairofthoseanglesequalhere.Youcanfollowthesameapproachtoprovetheotherthreepairsofanglesequal.

123 4

567 8

k

m

n

78

Given:LinemandLinekareparallelandintersectedbylinen

Prove:m2=m6


123 4

567 8

k

m

n

79

Statement1LinemandLinekareparallelandintersectedbylinen

Reason1Given


123 4

567 8

k

m

nGiven:LinemandLinekareparallelandintersectedbylinen

Prove:m2=m6

MathPractic

e

MP7Emphasizethatthe1ststeptoanyproofisstatingthe"Givens".Then,oneusesthepropertiesofthe1st

statementtoaskquestionsandcontinuetosolvethe

proof.

80

FifthPostulate:That,ifastraightlinefallingontwostraightlinesmaketheinterioranglesonthesamesidelessthantworightangles,thetwostraightlines,ifproducedindefinitely,meetonthatsideofwhicharetheangleslessthanthetworightangles.

RememberEuclid'sFifthPostulate.Theonethatnoonelikesbutwhichtheyneed.Thisiswhereit'sneeded.



123 4

567 8

k

m


Prove:m2=m6

81


1 2

3 4

Recallthatwelearnedearlyinthisunitthatthismeansthat...

Ifthepairsofinterioranglesonbothsidesofthetransversal,(both1&3or2&4)eachaddupto180,thetworedlinesareparallel...andnevermeet.

82

123 4

567 8

k

m

n

27 So,inthiscase,whichanglesmustaddupto180basedonEuclid'sFifthPostulate?

A 1&4

B 6&8

C 4&5

D 3&6

E Alloftheabove

Ans

wer

C&D

83

Reason2Euclid'sFifthPostulate

Statement2m3+m6=180m4+m5=180

Whenanglessumto180,whattypeofanglesarethey?


123 4

567 8

k

m

n


Prove:m2=m6

MathPractic

e

QuestionsonthisslideaddressMP2&MP3.

84

Reason3DefinitionofSupplementaryAngles

Statement33&6aresupplementary4&5aresupplementary

Whichotherangleissupplementaryto3,becausetogethertheyformastraightangle?Howabouttoangle6?


123 4

567 8

k

m

n


Prove:m2=m6

MathPractic

e


85

Reason4Anglesthatformalinearpairaresupplementary

Statement42&3aresupplementary

Whatdoweknowaboutangleswhohavethesamesupplements?


123 4

567 8

k

m

n


Prove:m2=m6

MathPractic

e


86

Reason5Twoanglessupplementarytothesameangleareequal

Statement5m2=m6


123 4

567 8

k

m

n


Prove:m2=m6

87

Statement Reason

1 LinemandLinekareparallelandintersectedbyLinen Given

2 m4+m5=180m3+m6=180


3 4&5aresupplementary3&6aresupplementaryDefinitionofsupplementaryangles

4 3&2aresupplementary Anglesthatformalinearpairaresupplementary

5 m2=m6Twoanglessupplementarytothesameangleareequal

CorrespondingAnglesProofGiven:LinemandLinekareparallelandintersectedbyLinen

Prove:m2=m6

123 4

567 8

k

m

n

88


Thisisanimportantresult,whichwasonlymadepossiblebyEuclid'sFifthPostulate.

Itleadstosomeotherprettyimportantresults.Itallowsustoprovesomepairsofanglescongruentandsomeotherpairsofanglessupplementary.

And,itworksinreverse,ifanyoftheseconditionsaremetwecanprovethatlinesareparallel.

123 4

567 8

k

m

n

89

ConversesofParallelLineProofs

Weprovedthatiftwolinesareparallel,theircorrespondinganglesareequal.

Theconversemustalsobetrue:

Iftwolinesarecutbyatransversalandthecorrespondinganglesarecongruent,thenthelinesareparallel. Te

ache

rNotes

Youmightneedtoexplainthedifferencebetweenanoriginalifthenstatementanditsconverse,whichisformedbyswitchingthehypothesis&conclusion.Ex:Original:Ifitis3pminNewJersey,thenitis1pminColorado.Converse:Ifitis1pminColorado,thenitis3pminNewJersey.

90


Thesamereason:CorrespondingAnglesofParallelLinesareEqualisusedineachcase.

Toprovetherelationshipbetweencertainanglesifweknowthelinesareparallel

OR

Toprovethatthelinesareparallelifweknowtherelationshipbetweenthoseangles.

91


Thispatternwillbetrueofeachtheoremweproveabouttheanglesformedbythetransversalintersectingtheparallellines.

Theyprovetherelationshipbetweenanglesoflinesknowntobeparallel,ortheyprovethatthelinesareparallel.

92

123 4

567 8

k

m

n

AlternateInteriorAnglesTheorem

AccordingtotheAlternateInteriorAnglesTheoremwhichoftheseanglesarecongruent?

Ifparallellinesarecutbyatransversal,thenthealternateinterioranglesarecongruent.

Ans

wer

93


Prove:35and46

AlternateInteriorAnglesProof

123 4

567 8

k

m

n

94

Statement1LinemandLinekareparallelandintersectedbylinen

Reason1Given


AccordingtotheCorrespondingAnglesTheoremwhichoftheaboveanglesarecongruent?

123 4

567 8

k

m

n


Prove:35and46

MathPractic

e

QuestiononthisslideaddressesMP3&MP6.

95

Reason2Whentwoparallellinesarecutbyatransversal,thecorrespondinganglesarecongruent.

Statement21526

Whichotherangleiscongruentto1?Whichotherangleiscongruentto2?Whyaretheseanglescongruent?


123 4

567 8

k

m


Prove:35and46

MathPractic

e

QuestionsonthisslideaddressMP3,MP6&MP7.

96

Reason3Verticalanglesarecongruent.

Whatdoweknowaboutanglesthatarecongruenttothesameangle?Explainyouranswer.


Statement31324

123 4

567 8

k

m

n


Prove:35and46

MathPractic

e

QuestionsonthisslideaddressMP2,MP3&MP6.

97

Reason4Transitivepropertyofcongruence

Statement4 3 5 4 6

Butthosearethepairsofalternateinteriorangleswhichwesetouttoprovearecongruent.So,ourproofiscomplete:AlternateInteriorAnglesofParallelLinesareCongruent


123 4

567 8

k

m

n


Prove:35and46

98

Statement Reason

1 LinemandLinekareparallelandintersectedbyLinen Given

2 15and26Iftwoparallellinesarecutbyatransversal,thenthecorrespondinganglesare

3 13and24 VerticalAnglesare

4 35and46TransitivePropertyofCongruence

AlternateInteriorAnglesProofGiven:LinemandLinekareparallelandintersectedbyLinen

Prove:3546

123 4

567 8

k

m

n

99

ConverseofAlternateInteriorAnglesTheorem

Iftwolinesarecutbyatransversalandthealternateinterioranglesarecongruent,thenthelinesareparallel.

100

123 4

567 8

k

m

n

AlternateExteriorAnglesTheorem

Iftwoparallellinesarecutbyatransversal,thenthealternateexterioranglesarecongruent.

AccordingtotheAlternateExteriorAnglesTheoremwhichanglesarecongruent?

Ans

wer

AccordingtoAlternateExteriorAnglesTheoremthe

followinganglesarecongruent:

1728

101

AlternateExteriorAnglesTheorem

SincetheprooffortheAlternateExteriorAnglesTheoremisverysimilartotheAlternateInteriorAnglesTheorem,youwillbecompletingthisproofasapartofyourHomeworkforthislesson.

102

ConverseofAlternateExteriorAnglesTheorem

Iftwolinesarecutbyatransversalandthealternateexterioranglesarecongruent,thenthelinesareparallel.

103

123 4

567 8

k

m

n

SameSideInteriorAnglesTheorem

Iftwoparallellinesarecutbyatransversal,thenthesamesideinterioranglesaresupplementary.

AccordingtotheSameSideInteriorAnglesTheoremwhichpairsofanglesaresupplementary?

Answ

er

Ans

wer

AccordingtoSameSideInteriorAnglesTheoremthe

followinganglesaresupplementary:

m3+m6=180m4+m5=180

104

SameSideInteriorAnglesProof

Given:Linesmandkareparallelandintersectedbylinen

Prove:3&6aresupplementaryand4&5aresupplementary

123 4

567 8

k

m

n

105

28 Whichreasonappliestostep1?

Statement Reason

1 Linesmandkareparallelandintersectedbylinen ?

2 m3+m6=180m4+m5=180 ?

3 ? Definitionofsupplementarys

123 4

567 8

k

m

nA DefinitionofsupplementaryB Euclid'sFifthPostulateC GivenD AlternateInterior sareE Corresponding sare A

nswer

C

106


A DefinitionofsupplementaryB Euclid'sFifthPostulateC GivenD AlternateInterior sareE Corresponding sare

Statement Reason


2 m3+m6=180m4+m5=180 ?

3 ? Definitionofsupplementarys

123 4

567 8

k

m

n

Ans

wer

107

30 Whichstatementshouldbeinstep3?

A B C D E

Statement Reason

1 LinemandLinekareparallelandintersectedbyLinen ?

2 Thesumsofm3andm6andofm4andm5are180. ?

3 ?Definitionofsupplementary

angles

123 4

567 8

k

n

m

3and6aresupplementary6and5aresupplementary2and6aresupplementary4and5aresupplementary3and5aresupplementary A

nswer

108

Statement Reason

1 Linesmandkareparallelandintersectedbylinen Given

2 m3+m6=180m4+m5=180 Euclid'sFifthPostulate

3 3and6aresupplementary4and5aresupplementary Definitionofsupplementarys

SameSideInteriorAnglesProof

Given:LinemandLinekareparallelandintersectedbyLinen

Prove:3&6aresupplementaryand4&5aresupplementary

123 4

567 8

k

m

n

109

ConverseofSameSideInteriorAnglesTheorem

Iftwolinesarecutbyatransversalandthesamesideinterioranglesaresupplementary,thenthelinesareparallel.

110

123 4

567 8

k

m

n

SameSideExteriorAnglesTheoremIftwoparallellinesarecutbyatransversal,thenthesamesideexterioranglesaresupplementary.

AccordingtotheSameSideExteriorAnglesTheoremwhichanglesaresupplementary?

Ans

wer

AccordingtoSameSideExteriorAnglesTheoremthefollowinganglesare

supplementary:

m2+m7=180m1+m8=180

111

SameSideExteriorAnglesProof

Given:LinesmandkareparallelandintersectedbyLinen

Prove:2&7aresupplementary

Inprovingthat2&7aresupplementarywearetherebyprovingthat1&8aresupplementaryasthesameargumentsapplytobothpairsofangles.

123 4

567 8

k

m

n

112

31 Whichreasonappliestostep1?A Definitionofsupplementary sB SubstitutionpropertyofequalityC GivenD sthatformalinearpairaresupplementary

Statement Reason


2 ? Samesideinterioranglesaresupplementary

3 ? Anglesthatformalinearpairaresupplementary4 26and37 ?5 2&7aresupplementary ?

E ssupplementarytothesame are

123 4

567 8

k

m

n

Ans

wer

C

113

Statement Reason




32 Whichstatementismadeinstep2?

A 2&1aresupplementaryB 7&8aresupplementaryC 3&6aresupplementaryD 4&5aresupplementaryE 5&8aresupplementary

123 4

567 8

k

m

n

Ans

wer

114

33 Whichstatementismadeinstep3?A B C D E

2&3aresupplementary1&3aresupplementary6&8aresupplementary6&7aresupplementary7&1aresupplementary

Statement Reason




123 4

567 8

k

m

n

Ans

wer

115

Statement Reason




34 Whichreasonappliestostep4?A Definitionofsupplementary sB SubstitutionpropertyofequalityC GivenD sthatformalinearpairaresupplementary

E ssupplementarytothesame are

123 4

567 8

k

m

n

Ans

wer

116


Statement Reason



3 ? Anglesthatformalinearpairaresupplementary4 26and37 ?

5 2&7aresupplementary ?

A DefinitionofsupplementarysB SubstitutionpropertyofequalityC GivenD Anglesthatformalinearpairaresupplementary

E ssupplementarytothesameare

123 4

567 8

k

m

n

Ans

wer

117

SameSideExteriorAnglesProof

Statement Reason

1 Linesmandkareparallelandintersectedbylinen Given

2 3&6aresupplementary Samesideinterioranglesaresupplementary

3 2&3aresupplementary6&7aresupplementaryAnglesthatformalinearpairaresupplementary

5 26and37Anglessupplementarytothesameanglearecongruent

6 2&7aresupplementary SubstitutionPropertyofEquality

Given:LinemandLinekareparallelandintersectedbyLinen

Prove:2&7aresupplementary(andtherebythat1&8areaswell)

123 4

567 8

k

m

n

118

ConverseofSameSideExteriorAnglesTheorem

Iftwolinesarecutbyatransversalandthesamesideexterioranglesaresupplementary,thenthelinesareparallel.

119



120


Thereareseveraltheoremsandpostulatesrelatedtoparallellines.Atthistime,pleasegotothelabtitled,"PropertiesofParallelLines".

Clickheretogotothelabtitled,"PropertiesofParallelLines"

MathPractic

e

ThislabaddressesMP1,MP3,MP4,MP5,MP6,MP7&MP8

https://njctl.org/courses/math/geometry/parallel-lines/attachments/properties-of-parallel-lines-2/https://njctl.org/courses/math/geometry/parallel-lines/properties-of-parallel-lines/

121


123 4

567 8

k

n km

m

Example:Ifm4=54,findthem8.

Explainyouranswer.

Ans

wer

AccordingtoCorrespondingAnglesPostulate48,therefore

m8=54.

ThisexampleaddressesMP1,MP2&MP3

122



Explainyouranswer.

123 4

567 8

k

n km

m

Ans

wer


123



Explainyouranswer.

123 4

567 8

k

m

n km

Ans

wer


124


Example:Ifm3=163,findm6.Explainyouranswer.

123 4

567 8

k

m

n km

Ans

wer


125


Namealloftheanglescongruentto1.

123 4

567 8

k

m

n km

Ans

wer

1357

126


Namealloftheanglessupplementaryto1.

123 4

567 8

k

m

n km

Ans

wer

1issupplementarytoangles:2,4,6,and8.

127

36 Findalloftheanglescongruentto5.A 1B 4C 8D alloftheabove

1 23 4

5 687

jm

j

m

k

Ans

wer

D

128

37 Findthevalueofx.

j

m(5x+30)

120

jm

k

Ans

wer

129

38 Findthevalueofx.

j

m(1.5x+40)

110

jmk

Ans

wer

130

39 Ifthem4=116thenm9=_____?

kmnp

n p

2 13 4

567 8

91011 12

131415 16

k

m

Ans

wer

64

131

40 Ifthem15=57,thenthem2=_____.

A 57B 123C 33D noneoftheabove

kmnp

n p

2 13 4

567 8

91011 12

131415 16

k

m

Ans

wer

132

Withthegivendiagram,notransversalexistsbutwecanextendoneofthelinestomakeatransversal.

ExtendingLinestoMakeTransversals

131

1

41

Findm1.

133

Thenfillintheanglewhichiscorrespondingtothe131angle.Whichanglecorrespondstothe131?


131

1

41

Findm1.

Ans

wer

ThequestiononthisslideaddressesMP7.

Thetopangleinthesetof4anglesinthefigure(ontherightsideofthefigure).

134

Thenfindthemeasurementoftheangleadjacentto131thatisinsideofthetriangle.Whatisthemeasurementofthisangle?Explainyouranswer.


131

1

41

131

Findm1.

Ans

wer

ThequestionsonthisslideaddressMP2&MP3.

49degreesTheanglesarealinearpair,

whichmakesthemsupplementary.

135

Asyoumayrecall,thethirdangleinthetrianglemustmakethesumoftheanglesequalto180.Whatisthemeasurementofthe3rdangleinthetriangle?


131

1

41

13149

Findm1.

Ans

wer


136

And,finallythatangle1issupplementarytothat90angle.Whatism1?


131

1

41

13149

90

Findm1.

MathPractic

e


137

m1=90


131

1

41

13149

90

Findm1.

138

Findthevaluesofxandy.

DoubleTransversals

132

x

(4y+12)

Ans

wer

AdditionalQ'sthataddressMP's:Whatinformationareyougiven?(MP1)Whatdoyouneedtofind?(MP1)Createanequationtorepresenttheproblem.(MP2)Howaretheanglesw/theexpressionsrelatedthe132angle?(MP7)

139

(14x+6) 66

2z

(3y6)

Findthevaluesofx,y,andz.

TransversalsandPerpendicularLines

Ans

wer

140

41 Findthem1.

1

126

110

Ans

wer

141

42 Findthevalueofx.

(3x)

54

A 12B 54C 42D 18

Ans

wer

142

43 Findthevalueofx.

(2x3)

(4x61) Ans

wer

143

122

(16x+10)

44 Findthevalueofx.

Ans

wer

144

Ifm3=56,findthem7thatmakeslineskandmparallel.

Explainyouranswer.

123 4

567 8

k

m

n

ProvingLinesareParallel

Ans

wer

AccordingtotheConverseoftheCorresponding

AnglesTheorem,ifm3=m7,thenkm.

Therefore,ifm3=56,thenm7=56.


145


Explainyouranswer.


Ans

wer

AccordingtotheConverseoftheAlternateInteriorAnglesTheorem,ifm4=

m6,thenkm.Therefore,ifm4=110,thenm6=110.


123 4

567 8

k

m

n

146


Explainyouranswer.


Ans

wer

AccordingtotheConverseoftheAlternateExteriorAnglesTheorem,ifm1=m7,thenkm.Ifm1=48,

thenm7=48


123 4

567 8

k

m

n

147


Explainyouranswer.


123 4

567 8

k

m

n

Ans

wer

AccordingtotheConverseoftheSameSideAnglesTheorem,ifm5+m4=

180,thenkm.Therefore,ifm5=54,thenm4=126 .


148

45 Whichstatementwouldshowlineskandmparallel?

123 4

567 8

k

m

n

A m2=m4B m5+m6=180

C m3=m5D m1+m5=90

Ans

wer

149

46 Inthisdiagram,whichofthefollowingistrue?

12364 57

132

e f g

h

i

A efB fgC hiD eg

Ans

wer

D

150

47 IflinesaandbarecutbyatransversalwhichofthefollowingwouldNOTprovethattheyareparallel?

A Correspondinganglesarecongruent.B Alternatneinterioranglesarecongruent.C Samesideinterioranglesarecomplementary.D Samesideinterioranglesaresupplementary.E Alloftheabove. An

swer

C

151

48 Findthevalueofxforwhichab.

a

bx

115

d

Ans

wer

152

49 Findthevalueofxwhichmakesab.

(6x20)

2x

a

b

cd

Ans

wer

x=25

153

50 Findthevalueofxforwhichmn.

m

n

(14x10)

(5x) Answ

er

10

154

51 Ifab,howcanweprovem1=m4?

A CorrespondinganglestheoremB ConverseofcorrespondinganglestheoremC AlternateInterioranglestheoremD Converseofalternateinterioranglestheorem

a b

c1 4

32

Ans

wer

C

155

52 Ifm1=m3,howcanweproveab?

A CorrespondinganglestheoremB ConverseofcorrespondinganglestheoremC AlternateInterioranglestheoremD Converseofalternateinterioranglestheorem

a b

c1 4

32

Ans

wer

156

53 Givenm1=m2,m3=m4,whatcanweprove?(chooseallthatapply)

A abB cdC lineaisperpendiculartolinecD linebisperpendiculartolined

ab

d

12

3

4 5 c

Ans

wer

157

54 Givenab,whatcanweprove?

A m1=m2B m1=m4C m2=m3D m1+m3=180

a b

c1 4

32

Ans

wer

158

ConstructingParallelLines


MathPractic

e

Thisentirelessonw/constructionsaddressesMP5

159

ParallelLineConstruction

Constructinggeometricfiguresmeansyouareconstructinglines,angles,andfigureswithbasictoolsaccurately.

Weuseacompass,andstraightedgeforconstructions,butwealsousesomepaperfoldingtechniques.

Clickheretoseeananimatedconstructionofaparallellinethrough

apoint.

Constructionby:MathIsFun

http://www.mathsisfun.com/geometry/construct-paranotline.html

160

Given:LineABandpointC,notontheline,drawasecondlinethatisparalleltoABandgoesthroughpointC.

Therearethreedifferentmethodstoachievethis.

Method1:CorrespondingAngles

ParallelLineConstruction

A

C

B

161

Thetheoryofthisconstructionisthatthecorrespondinganglesformedbyatransversalandparallellinesareequal.

Tousethistheory,wewilldrawatransversalthroughCthatcreatesanacuteanglewithlineAB.

ThenwewillcreateacongruentangleatC,onthesamesideofthetransversalastheacuteangleformedwithlineAB.

Sincethesearecongruentcorrespondingangles,thelinesareparallel.

A

C

B

ParallelLineConstruction:Method1

162

Step1:DrawatransversaltoABthroughpointCthatintersectsABatpointD.AnacuteanglewithpointDasavertexisformed(themeasureoftheangleisnotimportant).


A

C

BD

TheangleCDBistheanglewewillreplicateatpointConthesamesideofthetransversal.

163

A

C

BD

F

E

Step2:CenterthecompassatpointDanddrawanarcthatintersectsbothlines.Usingthesameradiusofthecompass,centeritatpointCanddrawanotherarc.LabelthepointofintersectiononthesecondarcF.

0135Wearefollowingthe

procedureweusedpreviouslytoconstructacongruentangle.

ThisstepistomarkthesamedistancesfromDandfromC.


164

Step3:Setthecompassradiustothedistancebetweenthetwointersectionpointsofthefirstarc.

A

C

BD

F

0

118

Thisreplicatesthedistancebetweenwherethearcintersectsthetwolegsoftheangleatthesamedistancefromthevertex.

WhenthatisreplicatedatCtheangleconstructedwillbecongruentwiththeoriginalangle.


165

Step4:CenterthecompassatthepointFwherethesecondarcintersectslineDCanddrawathirdarc.

A

C

BD

F

0

118


Thisassuresthatthearclengthforeachangleisidentical.

166

Step5:MarkthearcintersectionpointEanduseastraightedgetojoinCandE.

A

C

BD

F

E


CDB FCEthereforeABCE

167

Herearemyparallellineswithouttheconstructionlines.

A BD

C E


168

VideoDemonstratingConstructingParallelLineswithCorrespondingAngles

usingDynamicGeometricSoftware

Clickheretoseevideo

http://youtu.be/OZwIX9kVp0M

169

Thetheoryofthisconstructionisthatthealternateinterioranglesformedbyatransversalandparallellinesareequal.

Tousethistheory,wewilldrawatransversalthroughCthatcreatesanacuteanglewithlineAB.

ThenwewillcreateacongruentangleatC,ontheoppositesideofthetransversalastheacuteangleformedwithlineAB.

Sincethesearecongruentalternateinterioranglesthelinesareparallel.

A

C

B


170

A B

C

Method2:AlternateInteriorAnglesGivenABandpointC,notontheline,drawasecondlinethatisparalleltoABandgoesthroughpointC.

171

A B

C

D

Method2:AlternateInteriorAnglesStep1:DrawatransversaltolineABthroughpointCthatintersectslineABatpointD.AnacuteanglewithpointDasavertexisformed.

TheangleCDBistheanglewewillreplicateatpointContheoppositesideofthetransversal.

172

A B

C

D

F

E

Method2:AlternateInteriorAngles

0 148

Step2:CenterthecompassatpointDanddrawanarcthatintersectsbothlines,atpointsEandatF.Wearefollowingtheprocedureweusedpreviouslytoconstructacongruentangle.

ThisstepistomarkthesamedistancefromDonbothlines.

173

A B

C

D

F

E

G


0144

Step3:Usingthesameradius,centerthecompassatpointCanddrawanarcthatpassesthroughlineDCatpointG.

ThisreplicatesthesamedistancealongthetransversalandthenewlinethatwillbedrawnfromCaswasdoneforthedistancesfromD.

174

A B

C

D

F

E

G

H

Method2:AlternateInteriorAnglesStep4:Again,withthesameradius,centerthecompassatpointGanddrawathirdarcwhichintersectstheearlierone,atH.

0

144

Thisnowfindsthatsamedistancefromwherethearcintersectsthetransversalandthenewlineaswasthecaseforthetransversalandtheoriginalline.

175

A B

C

D

F

E

G

H

Step5:DrawlineCH,whichwillbeparalleltolineABsincetheiralternateinterioranglesarecongruent.


0

144

SinceanglesHCGandBDFarecongruentandarealternateinteriorangles,thelinesareparallel.

176

A B

C

D

H

Herearethelineswithouttheconstructionstepsshown.


177

VideoDemonstratingConstructingParallelLineswithAlternateInteriorAngles

usingDynamicGeometricSoftware

Clickheretoseevideo

http://youtu.be/p2iPjRX3LOM

178

A B

C

Method3:AlternateExteriorAnglesGivenlineABandpointC,notontheline,drawasecondlinethatisparalleltolineABandgoesthroughpointC.

179

B

C

DA

Method3:AlternateExteriorAnglesStep1:DrawatransversaltolineABthroughpointCthatintersectslineABatpointD.AnacuteanglewithpointDasavertexisformed.

180

A B

C

D

E

Method3:AlternateExteriorAnglesStep2:CenterthecompassatpointDanddrawanarctointersectlinesABandDContheoppositesideofpointCatAandE. 0

168

181

A B

C

D

F

E

Method3:AlternateExteriorAnglesStep3:KeepingtheradiusthesamedrawanarccenteredonCthatintersectslineDCaboveC,atF.

0

168

182

A B

C

D

F

E

G

Method3:AlternateExteriorAnglesStep4:StillkeepingtheradiusthesamedrawanarccenteredonFthatintersectsthearccenteredonC,atH.

0168

183

A B

C

D

F

E

G

Step5:DrawlineCE,whichisparalleltolineABsincethealternateexterioranglesformedbythetransversalarecongruent.

Method3:AlternateExteriorAngles

ADGECFthereforeABCE

184

A B

C E

Herearethelineswithouttheconstructionlines.Method3:AlternateExteriorAngles

185

VideoDemonstratingConstructingParallelLineswithAlternateExterior

AnglesusingDynamicGeometricSoftware

Clickheretoseevideo

http://youtu.be/GnZ5E7i3qaQ

186

ParallelLineConstructionUsingPattyPaper

Step1:Drawalineonyourpattypaper.Labelthelineg.DrawapointnotonlinegandlabelthepointB.

gB

187

gB


Step2:FoldyourpattypapersothatthetwopartsoflineglieexactlyontopofeachotherandpointBisinthecrease.

188

Step3:Openthepattypaperanddrawalineonthecrease.Labelthislineh.


gB

h

189

Step4:ThroughpointB,makeanotherfoldthatisperpendiculartolineh.


gB

h

190

Step5:Openthepattypaperanddrawalineonthecrease.Labelthislinei.

Becauselinesiandgareperpendiculartolinehtheyareparalleltoeachother.Thereforelineilineg.


gB

h

i

191

VideoDemonstratingConstructingaParallelLineusingMenuOptionsof

DynamicGeometricSoftware

Clickheretoseevideo2

Clickheretoseevideo1

http://youtu.be/sBJBWp-61l0http://youtu.be/KiURcVkRGx0

192

C

A B

E

D

F

G

55 Thelinesinthediagrambelowareparallelbecauseofthe:A AlternateInteriorAnglesTheoremB AlternateExteriorAnglesTheoremC SameSideAnglesTheoremD CorrespondingAnglesPostulate

Ans

wer

0109

193

56 Thelinesbelowareshownparallelbythe:

A AlternateInteriorAnglesTheoremB AlternateExteriorAnglesTheoremC SameSideAnglesTheoremD CorrespondingAnglesPostulate

C

A

E

D

F

G

Ans

wer

A

194

57 Thebelowlinesareshownparallelbythe:A AlternateInteriorAnglesTheoremB AlternateExteriorAnglesTheoremC SameSideAnglesTheoremD CorrespondingAnglesPostultate

C

AD

FG

E

Ans

wer

195

PARCCSampleTestQuestions

TheremainingslidesinthispresentationcontainquestionsfromthePARCCSampleTest.Afterfinishingunit3,youshouldbeabletoanswerthesequestions.

GoodLuck!


196


PARCCReleasedQuestion(EOY)

Topic:ParallelLines&Proofs

http://parcc.pearson.com/practice-tests/math/

197

58 CBDBFE

A GivenB DefinitionofcongruentanglesC VerticalanglesarecongruentD ReflexivepropertyofcongruenceE SymmetricpropertyofcongruenceF Transitivepropertyofcongruence

BA

D

E

F

G

C

H

Inthefigureshown,LineCFintersectslinesADandEHatpointsBandF,respectively.

Given:CBDBFEProve:ABFBFE

Ans

wer

A

198

59 CBDABF

A GivenB DefinitionofcongruentanglesC VerticalanglesarecongruentD ReflexivepropertyofcongruenceE SymmetricpropertyofcongruenceF Transitivepropertyofcongruence

BA

D

E

F

G

C

H



Ans

wer

C

199

60 ABFBFEA GivenB DefinitionofcongruentanglesC VerticalanglesarecongruentD ReflexivepropertyofcongruenceE SymmetricpropertyofcongruenceF Transitivepropertyofcongruence

BA

D

E

F

G

C

H



Ans

wer

F

200

BA

D

E

F

G

C

H



Completedproofshownbelow.

Statement Reason

1 CBDBFE Given

2 CBDABF VerticalAnglesarecongruent

3 ABFBFE Transitivepropertyofcongruence

201


Circlethereasonthatsupportseachlineoftheproof.


Topic:ParallelLines&Proofs

http://parcc.pearson.com/practice-tests/math/

202

61 mCBD=mBFEA GivenB AnglesthatformalinearpairaresupplementaryC AnglesthatareadjacentaresupplementaryD ReflexivepropertyofequalityE SubstitutionpropertyofequalityF Transitivepropertyofequality

BA

D

E

F

G

C

H

InthefigureshownLineCFintersectslinesADandEHatpointsBandF,respectively.

Given:mCBD=mBFEProve:mBFE+ mDBF=180

Ans

wer

A

203

62 mCBD+mDBF=180

BA

D

E

F

G

C

H



A GivenB AnglesthatformalinearpairaresupplementaryC AnglesthatareadjacentaresupplementaryD ReflexivepropertyofequalityE SubstitutionpropertyofequalityF Transitivepropertyofequality

Ans

wer

B

204

63 mBFE+mDBF=180

A GivenB AnglesthatformalinearpairaresupplementaryC AnglesthatareadjacentaresupplementaryD ReflexivepropertyofequalityE SubstitutionpropertyofequalityF Transitivepropertyofequality

BA

D

E

F

G

C

H



Ans

wer

E

205

BA

D

E

F

G

C

H


Given:mCBD=mBFEProve:mBFE+mDBF=180

Statement Reason

1 mCBD=mBFE Given

2 mCBD+mDBF=180Anglesthatformalinearpairaresupplementary

3 mBFE+mDBF=180 SubstitutionPropertyofEquality

206

64 PARTAConsiderthepartialConstructionofalineparalleltorthroughpointQ.whatwouldbethefinalstepintheconstruction?

A DrawalinethroughPandSB DrawalinethroughQandSC DrawalinethroughTandSD DrawalinethroughWandS

Thefigureshowsliner,pointsPandTonliner,andpointQnotonliner.AlsoshownisrayPQ.

rTP

Q

rTP

Q

W

S

Ans

wer


http://parcc.pearson.com/#

207

65 PARTBOncetheconstructioniscomplete,whichofthereasonslistedcontributetoprovingthevalidityoftheconstruction?

A Whentwolinesarecutbyatransversalandthecorrespondinganglesarecongruent,thelinesareparallel.

B Whentwolinesarecutbyatransversalandtheverticalanglesarecongruent,thelinesareparallel.

C Definitionofsegmentbisector.D Definitionofananglebisector.

Thefigureshowsliner,pointsPandTonliner,andpointQnotonliner.AlsoshownisrayPQ.

rTP

Q

rTP

Q

W

S

Ans

wer

A



208

66 Thediagramrepresentsaportionofasmallcity.MapleStreetandPineStreetrunexactlyeasttowest.Oakavenuerunsexactlynorthtosouth.Allofthestreetsremainstraight.

A BirchStreetandElmStreetintersectatrightangles.

B MapleStreetandPineStreetareparallel.

C Ifmoreofthemapisshown,ElmStreetandOakAvenuewillnotintersect.

D PineStreetintersectsbothBirchStreetandElmStreet.E OakAvenueandMapleStreetareperpendicular.

Question1/7

PARCCReleasedQuestion(PBA)

Topic:Lines:Intersecting,Parallel&Skew

Whichstatementsmustbetruebasedonlyonthegiveninformation?Selectallthatapply.

Ans

wer


Attachments

constructparanotline.webloc

[InternetShortcut]URL=http://www.mathsisfun.com/geometry/construct-paranotline.html

SMART Notebook

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parallel lines lines that intersect are in a common plane. euclid's fifth postulate 18 lines that...

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