unit 1 constructions & unknown angles lesson 1: when two lines, segments or rays intersect...
TRANSCRIPT
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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