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Unit1Constructions&UnknownAngles

Lesson1:ConstructanEquilateralTriangleOpeningExerciseJoeandMartyareintheparkplayingcatch.Tonyjoinsthem,andtheboyswanttostandsothatthedistancebetweenanytwoofthemisthesame.Wheredotheystand?Howdotheyfigurethisoutprecisely?Whattoolortoolscouldtheyuse?Example1YouwillneedacompassMargiehasthreecats.Shehasheardthatcatsinaroompositionthemselvesatanequaldistancefromoneanotherandwantstotestthattheory.MargienoticesthatSimon,hertabbycat,inthecenterofherbed(atS),whileJoJo,herSiamese,islyingonherdeskchair(atJ).Ifthetheoryistrue,wherewillshefindMack,hercalicocat?UsethescaledrawingofMargie’sroomshownbelow,alongwiththecompass,andplaceanMwhereMackwillbeifthetheoryistrue.Whatkindofshapehavethecatsformed?Bespecific!!!!

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Vocabulary

Define DiagramPoint

• alocation• namedwithacapitalletter

Line• onedimensional• goesonforeverinbothdirections

Segment• ameasurablepartofalinethatconsistsof

twoendpointsandallthepointsbetweenthem.

Ray• alinethathasoneendpointandgoeson

foreverinonedirection

Collinear• pointsthatlieontheSAMELINE

Plane• twodimensional• goesonforeverinALLdirection

Coplanar• pointsthatlieintheSAMEPLANE

Circle• thelocusofallpointsthatareequidistant

fromagivenpointcalledthecenter

Radius• asegmentconnectingthecenterofacircleto

anypointonthecircle

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Example2YouwillneedacompassCedarCityboaststwocityparksandisintheprocessofdesigningathird.Theplanningcommitteewouldlikeallthreeparkstobeequidistantfromoneanothertobetterservethecommunity.Asketchofthecityappearsbelow,withthecentersoftheexistingparkslabeledasA andB .IdentifytwopossiblelocationsforthethirdparkandlabelthemasC andD onthemap.Clearlyandpreciselylistthemathematicalstepsusedtodetermineeachofthetwopotentiallocations.

A •

B •

ResidentialAreaElementarySchool HighSchool

ResidentialAreaIndustrialArea

LightCommercial(grocery,drugstore,drycleaners,etc.)

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Example3Inthefollowingfigure,circleshavebeenconstructedsothattheendpointsofthediameterofeachcirclecoincidewiththeendpointsofeachsegmentoftheequilateraltriangle.a. WhatisspecialaboutpointsD,E,andF?Explain

howthiscanbeconfirmedwiththeuseofacompass.

b. DrawDE,EF,andFD.Whatkindoftrianglemust

ΔDEF be?c. WhatisspecialaboutthefourtriangleswithinΔABC ?d. HowmanytimesgreateristheareaofΔABC thantheareaofΔCDE ?

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HomeworkYouwillneedacompassandastraightedge1. ΔABC isshownbelow.Isitanequilateraltriangle?Justifyyourresponse.2. ConstructequilateraltriangleABCusinglinesegment AB asoneside.Writeaclear

setofstepsforthisconstruction.

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H1

H2

H3

H1

H2

Lesson2:ConstructanEquilateralTriangleIIOpeningExerciseYouwillneedacompassandastraightedgeTwohomesarebuiltonaplotofland.Bothhomeownershavedogs,andareinterestedinputtingupasmuchfencingaspossiblebetweentheirhomesontheland,butinawaythatkeepsthefenceequidistantfromeachhome.Useyourconstructiontoolstodeterminewherethefenceshouldgoontheplotofland.

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Example1YouwillneedacompassandastraightedgeUsingtheskillsyouhavepracticed,constructthreeequilateraltriangles,wherethefirstandsecondtrianglesshareacommonside,andthesecondandthirdtrianglesshareacommonside.Ifweweretocontinueusingthisproceduretoconstruct3moreequilateraltriangles,whatkindoffigurewouldwehaveformed?

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Example2YouwillneedacompassandastraightedgeAsaclass,wearegoingtoconstructahexagoninscribedinacircle,usingthepreviousexampleasaguide.Howcouldyouconnectthemarkingsdifferentlytomakeaninscribedequilateraltriangle?

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Postulates

Ingeometry,apostulate,oraxiom,isastatementthatdescribesafundamentalrelationshipbetweenthebasictermsofgeometry.Postulatesareacceptedastruewithoutproof.

Postulate DiagramThroughanytwopoints,thereisexactlyoneline.

Throughanythreenon-collinearpointsthereisexactlyoneplane.

Alinecontainsatleasttwopoints.

Aplanecontainsatleastthreenon-collinearpoints.

Iftwolinesintersect,thentheirintersectionisexactlyonepoint.

Iftwoplanesintersect,thentheirintersectionisaline.

In1-4,usethediagramshowntotheright:1. Howmanyplanesareshowninthefigure?2. HowmanyoftheplanescontainpointsFandE?3. Namefourpointsthatarecoplanar.4. ArepointsA,BandCcoplanar?Explain.

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HomeworkYouwillneedacompassandastraightedge1. Constructanequilateraltriangleinscribedinacircle.(Usethehexagonconstruction

inExample2asaguide!)

2. ConstructscalenetriangleABCusingthelengthsofthe3segmentsshownbelow:

A B

B C

CA

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Lesson3:ConstructaPerpendicularBisectorOpeningExerciseCompareyourhomeworkanswerswithyourpartner.ShownbelowarethesegmentsfromQuestion#2.Ifneeded,usethespaceprovidedtoredrawscalenetriangleABC.

A B

B C

CA

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Vocabulary

Define DiagramAngle

• Theunionoftwonon-collinearrayswiththesameendpoint.

Degree• 1/360ofacircle

ZeroAngle• Arayandmeasures 0°

StraightAngle• Alineandmeasures180°

LinearPair• Apairofadjacentangleswhosenon-common

sidesareoppositerays(supplementalangles)

RightAngle• ananglethatmeasures

Perpendicular• whentwolines,segmentsorraysintersect

forminga angle

Equidistant• apointissaidtobeequidistantwhenitisan

equaldistancefromtwoormorethings

Midpoint• Apointthatishalfwaybetweenthe

endpointsofasegment

AngleBisector• Araythatdividesanangleintotwoequal

angles

SegmentBisector• Asegment,lineorraythatdividesasegment

intotwoequalsegments

90°

90°

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Define:PerpendicularBisectorTheperpendicularbisectorofsegmentABistheline_______________________________toABandpassesthroughthe_______________________________ofAB.Example1YouwillneedacompassandastraightedgeThinkingbacktothefenceprobleminLesson2,experimentwithyourconstructiontoolstodeterminetheproceduretoconstructaperpendicularbisector.Preciselydescribethestepsyoutooktobisectthesegment.

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Example2YouwillneedacompassUsingyourcompass,examinethefollowingpairsofsegments:

a. AC,BC

b. AD,BD

c. AE,BEBasedonyourfindings,fillintheobservationbelow:Anypointontheperpendicularbisectorofalinesegmentis_________________________________fromtheendpointsofthelinesegment.

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Example3YouwillneedacompassandastraightedgeConstructtheperpendicularbisectorsofAB,BCandCAonthetrianglebelow.Whatdoyounoticeaboutthesegmentsyouhaveconstructed?

A

B C

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Homework1. Howmanypointsdeterminealine?2. Howmanypointsdetermineaplane?Whatmustbetrueaboutthepoints?3. Twonon-parallellinesintersecthowmanytimes?4. Theintersectionoftwoplanesiswhatkindoffigure?5. Usingacompassandstraightedge,constructthefollowing: a. EquilateralTriangle b. PerpendicularBisector c. AHexagonInscribedinaCircle

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A

A

B C

D

Lesson4:ConstructaPerpendicularBisectorIIOpeningExerciseYouwillneedacompassandastraightedgeYouknowhowtoconstructtheperpendicularbisectorofasegment.Nowyouwillinvestigatehowtoconstructaperpendiculartoaline fromapointAnoton .Thinkabouthowyouhaveusedcirclesinconstructionssofarandwhytheperpendicularbisectorconstructionworksthewayitdoes.Thefirststepoftheinstructionshasbeenprovidedforyou.Discovertheconstructionandwritetheremainingsteps.

Steps:1. DrawacircleAsothatthecircleintersectslinelintwopoints.LabelthesepointsB

andC.

l l

l

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A

A

B C

D

Example1YouwillneedacompassandastraightedgeYouaregoingtousetheconceptofconstructingperpendicularlinestoconstructparallellines! Step1:Constructalineperpendicularto throughpointA.Callthisnewline . Step2:Constructalineperpendicularto throughpointA.Callthisnewline .Vocabulary

Define DiagramParallelLines

l1 l3

l3 l2

l1

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Example2YouwillneedacompassandastraightedgeWhatarethecharacteristicsofasquare?Usingthesecharacteristicsandyourknowledgeofconstructions,wearegoingtoconstructasquare.

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HomeworkYouwillneedacompassandastraightedgeAnisoscelestriangleisatrianglethathastwocongruentsides.Anisoscelesrighttrianglehasarightanglebetweenthetwocongruentsides(thesesidesareperpendicular!).Usingthesegmentbelowasoneofyourcongruentsides,constructanisoscelesrighttriangle.Listyourstepsbelow:Steps:

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Lesson5:CopyandBisectanAngleOpeningExerciseYouwillneedacompassandastraightedgeGiven∠ABC picturedbelow.a. Draw AC .b. Constructtheperpendicularbisectorof AC .c. Whatistherelationshipbetweentheperpendicularbisectorandthegivenangle?d. Doyouthinkthiswillalwaysbethecase?Explain.

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Example1YouwillneedacompassandastraightedgeConstructtheanglebisectorofthefollowingangleandlistthestepsoftheconstruction.ExercisesUsingthestepsyoulistedabove,bisecttheanglesshownbelow.

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Example2YouwillneedacompassandastraightedgeWearenowgoingtocopyanangle.Constructthenewangleandlistthestepsoftheconstruction.

ExercisesUsingthestepsyoulistedabove,copytheangleshownbelow.

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HomeworkUsingyourcompassandstraightedge,bisecteachanglebelow:1. 2.

3. 4.

Usingyourcompassandstraightedge,copytheanglebelow:5.

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Lesson6:PointsofConcurrenciesOpeningExerciseHoldthetransparencyoveryourhomework.Didyouranglesandbisectorscoincideperfectly?Usethefollowingrubrictoevaluateyourhomework,markingwhichareasapplytoyou:

NeedsImprovement Satisfactory Excellent

Fewconstructionarcsvisible Someconstructionarcsvisible Constructionarcsvisibleandappropriate

Fewverticesorrelevantintersectionslabeled

Mostverticesandrelevantintersectionslabeled

Allverticesandrelevantintersectionslabeled

Linesdrawnwithoutstraightedgeornotdrawn

correctly

Mostlinesneatlydrawnwithstraightedge

Linesneatlydrawnwithstraightedge

Fewerthan3anglebisectorsconstructedcorrectly

3ofthe4anglebisectorsconstructedcorrectly

Anglebisectorconstructedcorrectly

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A

A

B C

D

Example1YouwillneedacompassandastraightedgeWearenowgoingtolookatasecondwayofconstructingparallellinesusinganotherconstructionwearefamiliarwith…copyinganangle!Wewillwalkthroughtheprocedureasaclass,writingdownourstepsaswego.ConstructalineparalleltothegivenlinegoingthroughpointA:

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Example2Sketchtheperpendicularbisectorforeachsideofthetrianglepicturedbelow.Vocabulary

• When3ormorelinesintersectinasinglepointtheyare____________________________.

• Thispointofintersectioniscalledthe______________________________________________.

• Thepointofintersectionfor3perpendicularbisectorsiscalledthe___________________.Wewillusehttp://www.mathopenref.com/trianglecircumcenter.htmltoexplorewhathappenswhenthetriangleisrightorobtuse.Sketchthelocationofthecircumcenteronthetrianglesbelow:

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Example3Usingthepicturetotheright,marktherightanglesandcongruentsegmentsifpointPisthecircumcenterofΔABC .LookbackatExample3fromLesson4.Wediscoveredthat:Anypointontheperpendicularbisectorofalinesegmentis

_________________________________fromtheendpointsofthelinesegment.Basedonthisdiscoverywhatcouldyouconcludeabout:

APandBP?

BPandCP? CPandAP?Thistellsusthattheperpendicularbisectorswillalwaysbeconcurrent!

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Example4Sketchtheanglebisectorsforeachangleofthetrianglepicturedbelow.Vocabulary

• Thepointofintersectionfor3anglebisectorsiscalledthe___________________.Wewillusehttp://www.mathopenref.com/triangleincenter.htmltoexplorewhathappenswhenthetriangleisrightorobtuse.Sketchthelocationoftheincenteronthetrianglesbelow:

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Lesson7:SolveforUnknownAngles–AnglesandLinesataPointOpeningExerciseFillinthe“Fact/Discovery”columnbasedongeometryfactsyouhavelearnedinthepast!

Name Diagram Fact/Discovery

VerticalAngles

Anglesformingarightangle

AnglesonaLine

AnglesataPoint

Define: Complementary: Supplementary: Adjacent:

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Example1Findthemeasureofeachlabeledangle.Giveareasonforyoursolution.

Angle AngleMeasure Reason

∠1

∠2

∠3

∠4

∠5

Example2Usethefollowingdiagrampicturedtotherighttoanswerthefollowing:a. Nameananglesupplementaryto∠HZJ andprovide

thereason.

b. Nameananglecomplementaryto∠HZJ andprovidethereason.

c. Nameananglecongruentto∠HZJ andprovidethereason.

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ExercisesFindthevalueofxand/oryineachdiagrambelow.Showallstepstoyoursolution.1. 2. 3. 4.

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5.6.7.8.

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HomeworkInthefiguresbelow,ABandCDarestraightlines.Findthevalueofxand/oryineachdiagrambelow.Showallstepstoyoursolution.1.

2.3 4.

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Lesson8:SolveforUnknownAngles–TransversalsOpeningExerciseWithyourpartner,identifythefollowing:

InteriorAngles

ExteriorAngles

AlternateInteriorAngles

AlternateExteriorAngles

CorrespondingAngles

SameSideInteriorAngles

Propertiesofparallellinescutbytransversal:

• Thealternateinterioranglesarecongruent.• Thecorrespondinganglesarecongruent.• Thesamesideinterioranglesaresupplementary.

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Example1If m P n andm∠1 = 150° ,findthemeasureoftheremaininganglesandprovideyourreasoning.

Example2Ifm P n,findthevalueofxfortheproblemspicturedbelow.

AngleMeasure Reasoningm∠2 m∠3 m∠4 m∠5 m∠6 m∠7 m∠8

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ExercisesFindthevaluesofthemissing(labeled)anglesineachdiagrambelow.Showallstepstoyoursolutions.1. 2. 3. 4.

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HomeworkIn1-3,AB P CDandareintersectedbytransversalEFatGandH.1. Ifm∠EGB = 40° ,findtheremainingangles.2. If∠AGH = x + 40 and∠CHG = 3x + 60 ,findx.3. If∠AGH = 3x −10 and∠DHG = 7x − 42 ,find∠DHG .

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In4-5,findthemeasureofthemissing(labeled)angles.Showallwork!4. 5.

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Lesson9:SolveforUnknownAngles–AuxiliaryLinesOpeningExerciseUsingyourknowledgeofparallellines,answerthefollowing:1. Whatisthemeasureof∠ABC?2. Ifm∠1=16x-8,m∠2=4(y+8),andm∠3=14x+2,findxandy.

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Vocabulary

Define DiagramAuxiliaryLine

ExercisesUseauxiliarylinestofindtheunknown(labeled)angles.Showallwork!1. 2.

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Name Diagram Fact/Discovery

∠ SumofΔ

ExercisesFindthemeasureofthemissinglabeledangles.1. 2.

3. 4.

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Exercises5. Thedegreemeasuresoftheanglesofatrianglearerepresentedbyx,3xand5x–54.

Findthevalueofx.6. InΔABC ,themeasureof∠B is21lessthanfourtimesthemeasureof∠A ,andthe

measureof∠C is1morethanfivetimesthemeasureof∠A .Findthemeasure,indegrees,ofeachangleofΔABC .

7. InΔABC ,themeasureof∠ABC is x2 ,themeasureof∠BCA is−6x +100 ,andthe

measureof∠CAB is x + 56 .Findthemeasureof∠ABC .

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HomeworkFindthemeasureofthemissinglabeledangles.1. 2. 3. 4. 5. InΔABC ,themeasureof∠A is3lessthantwotimesthemeasureof∠C andthe

measureof∠B is11morethanthemeasureof∠C .FindthemeasureofeachangleofΔABC .

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Lesson10:SolveforUnknownAngles–AnglesinaTriangleOpeningExerciseFindx:Fillinthe“Fact/Discovery”columnbasedongeometryfactsyouhavelearnedinthepast!

Name Diagram Fact/Discovery

∠ SumofRightΔ

Exterior∠ ’sofaΔ

Base∠ ’sofIsoscelesΔ

EquilateralTriangle

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ExercisesIneachfigure,determinethemeasureoftheunknown(labeled)angles.Showyourwork!!!1. 2. 3. 4. 5. 6.

1

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Exercises7. InΔABC ,themeasureofangleBisthreetimesaslargeasangleA.Anexterior

angleatCmeasures140° .FindthemeasureofangleA.8. InΔCAT ,sideCT isextendedthroughTtoS.If∠CAT = x2 −3x ,∠ACT = 6x + 20 ,

and∠ATS = 2x2 − 5x ,findx.9. InisoscelestriangleABC,thevertexangleCis20morethantwicethebaseangles.

Findthemeasureofalltheanglesofthistriangle.10. InΔDEF ,∠D isarightangleand∠F is12degreeslessthantwicethemeasureof

∠E .Findm∠F

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ExercisesAfewmorechallengingquestions!Ineachfigure,determinethemeasureoftheunknown(labeled)angles.11. 12. 13.

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HomeworkInquestions1&2,findthemeasureofthemissinglabeledangle.1. 2.

Inquestions3&4,solveforx.3. 4.

5. InthediagrambelowofΔACD ,Bisapointon AC suchthatΔADB isanequilateraltriangle,andΔDBC isanisoscelestrianglewithDB ≅ BC .Findm∠C .

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Lesson11:UnknownAngleProofs–WritingProofsOpeningExerciseSolvethefollowingequationforx,showingeverystepinthesolvingprocess!!!

6x −12 = 4x + 2

Nowasaclasswearegoingtosolvethissameproblemasaformalproof.Given:6x −12 = 4x + 2 Prove:x=7WhyProofs?Oneofthemaingoalsinstudyinggeometryistodevelopyourabilitytoreasoncritically:todrawvalidconclusionsbaseduponobservationsandprovenfacts.Masterdetectivesdothissortofthingallthetime.TakealookasSherlockHolmesusesseeminglyinsignificantobservationstodrawamazingconclusions.http://www.youtube.com/watch?v=o30UY_flFgM&feature=youtu.beCouldyoufollowSherlockHolmes’reasoningashedescribedhisthoughtprocess?Inaproofwearegoingtouseknownfactstoendupwithanewlyprovenfact.Thisiscalleddeductivereasoning.

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Example1Giventhediagrampicturedtotheright,find:

a. m∠1 b. m∠2

Nowlet’sprovewhytheexteriorangleofatriangleisequaltothesumoftheremoteinteriorangles!Wouldthisrulechangeif∠ACB wasacute?

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Exercises1. Giventhediagrampicturedtotheright,provethat

verticalanglesarecongruent. Statements Reasons

2. Giventhediagrampicturedtotheright,provethat ∠w + ∠x + ∠z = 180° . Statements Reasons

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3. Giventhediagrampicturedtotheright,provethat∠w = ∠y + ∠z .

Statements Reasons 4. Giventhediagrampicturedtotheright,provethat

thesumoftheanglesmarkedbythearrowsis 360° .

Statements Reasons

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x

z y

Homework1. Giventhediagrampicturedtotheright,provethat ∠x + ∠y + ∠z = 180° .

Statements Reasons2. Giventhediagrampicturedtotheright,findthe

valuesofallthemissingangles.

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Lesson12:UnknownAngleProofs–ProofswithConstructionsOpeningExerciseIneachfigure,determinethemeasureoftheunknown(labeled)angles.1. 2.

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Example1Inthefiguretotheright, AB P DE and BC P EF .Provethat∠B ≅ ∠E .(Hint:extend BC andED .)

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Example2Intheidenticaldiagramspicturedbelow,thereareatleast2possibilitiesforauxiliarylines.Canyoufindthem?Giventhat AB PCD ,wearegoingtoprovez=x+y.Halfoftheclassisgoingtoprovethisusingthefirstauxiliaryline,whiletheotherhalfisgoingtoprovethisusingthesecondauxiliaryline.

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Example3Inthefiguretotheright, AB PCD and BC P DE .Provethat∠B +∠D = 180° .

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Homework1. Provethetheorem“Whenparallellinesarecut

byatransversal,alternateexterioranglesarecongruent”.(Thismeansyouareproving

usingotherpropertiesthatyouhavelearned.)

2. Inthefiguretotheright, AB P DE and BC P EF

Provethat∠ABC ≅ ∠DEF .

∠1 ≅ ∠2

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Lesson13:UnknownAngleProofs–ProofsofKnownFactsOpeningExerciseWeprovedverticalanglesarecongruentinLesson11andweknowthatifatransversalintersectstwoparallellinesthatthealternateinterioranglesarecongruent.Usingtheseknownfacts,provethecorrespondinganglesarecongruent.Vocabulary

• Proof:anexplanationofhowamathematicalstatementfollowslogicallyfromotherknownstatements.

• Theorem:amathematicalstatementthatcanbeproven;typicallywrittenintheform“if(hypothesis)-then(conclusion)”.

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Exercise1Onceatheoremhasbeenproved,itcanbeaddedtoourlistofknownfactandusedinproofsofothertheorems!Wenowhaveavailablethefollowingfacts:

• Verticalanglesarecongruent.(vert.∠s )• Alternateinterioranglesarecongruent.(alt.int.∠s , AB PCD )• Correspondinganglesarecongruent.(corr.∠s , AB PCD )

Wearegoingtoproveonemoreinthehomework:

• Interioranglesonthesamesideofthetransversalaresupplementary.(int.∠s ,

AB PCD )Useanyofthesefourfactstoprovethatthethreeanglesofatrianglesumto180°.Forthisproof,youwillneedtodrawanauxiliaryline,paralleltooneofthetriangle’ssidesandpassingthroughthevertexoppositethatside.Addanynecessarylabelsandwriteoutyourproof.

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Exercise2Eachofthethreeparallellinetheoremshasaconverse(orreversing)theoremasfollows:

Original Converse

Iftwoparallellinesarecutbyatransversal,thenalternateinterioranglesarecongruent.

Iftwolinesarecutbyatransversalsuchthatalternateinterioranglesarecongruent,thenthelinesareparallel.

Iftwoparallellinesarecutbyatransversal,thencorrespondinganglesarecongruent.

Iftwolinesarecutbyatransversalsuchthatcorrespondinganglesarecongruent,thenthelinesareparallel.

Iftwoparallellinesarecutbyatransversal,theninterioranglesonthesamesideofthetransversaladdto180°.

Iftwolinesarecutbyatransversalsuchthatinterioranglesonthesamesideofthetransversaladdto180°,thenthelinesareparallel.

Inthefigureattheright,∠1 ≅ ∠2 .Provethat AB PCD .

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Exercise3Constructaproofdesignedtodemonstratethefollowing:

Iftwolinesareperpendiculartothesameline,theyareparalleltoeachother.(a)Drawandlabeladiagram.(b)Statethegivenfactsandtheconjecturetobeproved.(c)Writeoutaclearstatementofyourreasoningtojustifyeachstep.

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Homework1. Prove: Whenparallellinesarecutbyatransversal,thesamesideinterior

anglesaresupplementary.2. Given:QuadrilateralABCD

∠C and∠D aresupplementary. ∠B ≅ ∠D

Prove: AB PCD