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10thMathSumAngles
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Allright.Welcomebackguys.Grabacalculatorifyouneed...Sotodaywearegoing
tobestartingthemeasurementlesson,measurementone.Wehavefourdays
basicallyofthis,andwe'redonewiththeentiresemester.Sotakealook,seewhat
yougot,we'lltalkaboutit.
Pledgeallegiance,totheflag,oftheUnitedStatesofAmerica.Andtotherepublic,
forwhichitstands,onenation,undergod,indivisible,withlibertyandjusticefor
all.
Youwantmetoputthatontheside?Yourguitar.
Uh,yeah.
Juststickitover,likeonthatshelfoverthere.
OhStudentisgoingtobethere.
Hmm?(tostudent)Yeahgositbackup,byStudent.
Okay,soyoucanusethat.(tostudent)Usethatone.
Allright,I'llgiveyouacoupleminutes,findthosemissingangles,grabacalculatorif
youneed.
Here.I'llputthisdownforyou.
Here,justusethisone.Whenwedoouractivity.
Student.Student.
Oh,didn'teventiethatone.Verygood.
Rememberthisone,Student?Insideananglesontriangle.
No.
Let'stakeanotherminute.
Whatkindoftriangleisthis?(tostudent)Rememberwhatthat'scalled?Isosceles.
Whatdoweknowaboutthebottomtoanglesinanisoscelestriangle?
They'rethesame.
Thereyougo.They'rethesame.There'syourhint.
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Isthisanisosceles?
Whywouldyousayit'sisosceles?
Becauseyoujusttoldheritisosceles.
(laughs)Whywouldthediagramshowusthatit'sisosceles?
Becausethelinesare...Idon'tknowtheword.Congruent.
Yes,buthowdoweknowthey'recongruent?
Becauseofthelittletickmarks.
Yeah,thelittletickmarks.
Butyoujustsaiditwasthetickmarks.
Yes,butthetickmarks,right.
Canwejustsubtract34from180andthedivideby2?
That'sa[inaudible00:04:23]strategy.
Youshouldgiveuslikethirtymoreminutes.
(laughs)We'lldoalittlemore,we'lldoalittlemore.Youdon'thavetodrawthe
triangles,butI'ddefinitelyappreciateit.(laughs)Sure.
Allright,sowhatkindoftriangleisthis?(tostudent)
Good.
Allright,yougotalesson?(tostudent)
Sowhatdoesithavetobe34?Dothesetwoangleshavetobethesame?(to
student)Don'thavetobe.Thesetwohavetobethesame.Solet'sthinkaboutit,
whatdoalltheanglesaddupto?Insideatriangle,whatdoeseveryangleaddup
to?Whatdegreesdotheyaddupto?Thereyougo.So,let'strytofigureoutwhat
bothofthesearetogether,yeah?Allright,let'sseewhatyougot.
Allright,I'llgiveyouguysalittlebitmoretime.Iseesomegoodanswershere,some
gooddiagrams.Excellent.Good.Yougotit?Easy.Allright.
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Yeah,whatdoyouneedStudent?
Arewegettingthetestsback?
Ihavenotgradedthemyet.(laughs)Iwasverybusyyesterday,withtheseguys
apparently.Idon't...Ifyouwanttoseeafterclass,Icantalkaboutitrealquick.
WhereamIsigning?
Whereitsaysmath.
Thisone?
Thankyou.
Allright.Allright,solet'stalkaboutit.Thisissomethingwe'vedonequiteawhile
ago,weprovedthis,what?Aquarterortwoago?Interioranglesofatriangle.
Today,andIknowitdoesn'tsayitonthetitle,we'regoingtobeextendingthisinto
polygons,whathappensifwehaveafoursidedshape,orafivesidedshape,ora
tensidedshape,orathousandsidedshape,right?We'regoingtosortof
extrapolatefromthereandfigureout,whataretheinterioranglesgongtoaddup
to.Andthenofcourse,justlikeyoudidforthese,findthosemissingpieces.Right?
Allright,soforthefirstguy,whatkindoftriangleisthis?Isosceles,ofcourse.Which
meansthesetwoanglesare?Congruent,sure,thesame.Allrightsowecanlabel
thisXdegreesifwereallywantedto.NowIknowmanyofyoudon'tneedtowrite
thefullequation,buttobeprecise,let'swriteitout.Weknowthesumofthe
interiorangles,isalwaysthesame,whichis?180.
Allright,sowecantechnicallywritethiswholeequation,andwhetheryouactually
didthisornot,thisisthethoughtthatyouguyshadwhenyouweredoingthis.So
basicallywehave34+X+X=that180.Andofcourse,whatyouguysactuallydidis,
youminus34andthenyoudividedby2.Sointheend,whenwedidthat,whatdid
wecomeoutwith?Seventy-threedegrees.
Nowbecausethisisanequation,oneofthegreatfeaturesthatwecanexploitis,
wecancheckouranswer.Soifyouhave73,orsomethingelse,whatifyouhave72,
oryougotadifferentanswer?Youcanaddthemup.Soyoucanaddup73,and73,
and34andchecktoseeifthatis180.Nowagain,thisone'sfairlystraightforward,
soprobablynotnecessarybutthatisawayofcheckingsuchasonatestjustto
makesurethatyougotitright.
Allright,secondone,righttriangle,sameidea.Wehaveour...we'llfitthisonhere.
64,plus90,plussomethingequals180.AndStudent,whatdidthatactuallycome
outto?Twenty-sixdegrees.Soundsaboutright.Okayso,fairlystraightforward.It's
basicarithmetic,sometimesyouhavetobecleverbecausecertainanglesare
congruentandtheydon'tgiveyouoneofthoseangles,becausethatwouldjust
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giveyoutheanswer,soyou'reequationneedstobeyouknow,alittlebitmore,
morecomplicated.But,ingeneral,it'ssortofadditionandsubtraction.Wewillget
ontosomeofthoseequationslateron,whereit'slike64+Xoryouknow,2X-3,and
youhavetomakethosebiggerequations.Andwekindoftalkedaboutthatinour
reviewlastclass.Butforthemostpart,alotofthemathyou'regoingtodotoday,
issimplyadditionandsubtraction.Sonotabigdeal.Itshouldbeprettygood.
Nohomeworklasttime,yeah?Allright,Ihavenotgradedyourtestsyet.Iwilldo
thoseassoonaspossible,ifyoureallywanttokindofgetanideaofwhatyouhave,
comeseemeafterclasstoday,wecangothroughitandIcan...Idon'tknow,I
don'tthinkIcangradeitthatquicklybutIcan...Wecantalkaboutitandseewhat
youguysactuallygot.I'lltryanddothatthisweekend,assoonaspossible.
Okaysotodaywe'regoingtobeinvestigatingandthenwritingdownourlast
theorem,whichhastodowiththeinterioranglesofanyshape.Notjustatriangle,
butwhataboutafoursided,fivesides,sixsidedshape,okay?We'regonnastart
withalittleactivity,sograbyourbooks,opentoablankpage,whichIguessisthe
nextpage.GraboneoftheprotractorsthatIputonyourdesk.Question.[inaudible
00:10:17]Yeahjustdoitonthere,that'sfine.
Allrightso,we'regoingtoinvestigateandifyouactuallylookinsideyourworkbook
onpage149there'salittletable,we'regonnafillthatoutlatersomaybeyouwant
tokeepthatofftotheside,itlookslikethisbytheway.Canwedoublecheck?Isit
159?WishIhadabookwithme.Thereisoneon150butIthinkthere'sanother
oneon149.Check149.Isthattheone?No,thereyougo.Okay,yes.There'salotof
tableshere,we'renotgoingtofillthemallout,there'sbasicallytwoideaswe're
goingtodo.Firstofall,we'regoingtoinvestigate.Andthisissortoftheinductive
reasoning.We'regoingtophysicallydrawthesepolygons,we'regoingtomeasure
theirangles,whichagainisnotthemostaccuratewayofdoingitbecausewe're
justgoingtomakethemup,andwe'regoingtoaddupthoseangles.We'regonna
trytosortofcomeupwithanidea,comeupwithatheoryabouthowtheseangles
arerelated,insideabunchofdifferentpolygons.
Nowthebookwantedyoutodoallofthemandobviouslythat'sgoingtotake
forever,soIjustbrokeyouguysupbytwopolygons.AndI'vesortofalternated
themsoyou'renotdoingamilliondifferentsides.Soonyourtablethereshouldbe
astickynote,it'llsaylikepolygonwithnineandfivesides,orsevenandfoursides,
orwhatever.Justtwodifferentpolygons,andsowhatyouneedtodo,isjustlikeI
didoverhere,makeapolygonwiththatnumberofsides.Andthenmakethe
secondpolygon,withthatnumberofsides.Justmakethemstraight,usethe
straightedge,theydon'thavetobeperfect,theydon'thavetoberegular,they
don'thavetohaveanyparticularlength.Butjustmakethemandthenas
accuratelyasyoucan,butroundedtothenearestwholenumber,pleasemeasure
thoseanglesandaddthemup.Okay.Sowe'lltakeacoupleminuteshere,seewhat
youguyscando.Youcanworkwiththepersonatyourtable,butdrawthem,
measurethem,andthencompare.Seeifyouguysgetthesamemeasurementsfor
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eachoftheseitems.Allright,takeacoupleminutes.
Doyouneedanything?(tostudent)Sure,grabyourthings.
They'renotgoingtocomeoutperfect.Butgetthemcloseenough.
Yeah?
CanIdoitonthispage?
Yeahthat'sfine.
Makesureyouguyshaveeightandsix,okay.
Whatisthatafourth?Sothat'swhenwe'regoingtobeconstructing,soifyouwant
todoitinyourbook,you'regoingtobedrawingthatshapeandthenmeasuring
thoseangles.Soforyou,youcandothesevenandfive.Drawasevensidedfigure,
aniceone.Andthenafivesidedfigure,aniceone.Youcandothetenandthe
four,sojustdrawa...not,again,notaperfectone.Butasniceasyoucan,sowe
canmeasuresharpangles.
Soyouhaveyour,whichoneisthis?Sixsides,okay.Sonowyouneedtomeasure
eachoftheseinterioranglesascloseasyoucan.Whatyouwanttodoisputthat
onthecorner,andlineitupasmuchaspossible.Andthenyouwanttoseewhere
doesthisthingyhit.Again,you'remeasuringthisangle.Soyou'regoingtostart
fromthissideandgoup,sothat'llbelike130.
Yeahpleasebecarefulguys,theprotractorhastwosidestoit.Oneofthem
countingup,oneofthemcountingdown.Lookattheangle,seeifit'sacuteor
obtuse,andthat'llbeaneasywaytoswitchtotellwhichsideyoushouldbeusing.
Justbecarefulwiththat,becauseIswitchthatupallthetimetoo.
Youguysgotyourshapes?Lookingcool,allright.Nowwe'regoingtomeasureeach
oftheseangles.Getthefiveoneofthese,that'stotallyfine.I'llgiveyouoneof
these,it'skindofhardtomeasurewhenwemakethemsosmall.Butbasically...
Here,putthatonhereagain.Ifyouhavethis,youlineitupwithoneofthesides
andthenyousortofsee,wheredoestheothersideendup.Soyoucanseehere,
whatdegreeisthat?90isrighthere,wegottocuntupalittlebit.93,perfect.
Ascloseasyoucan,I'mnotsurehowaccuratethisisgoingtobebut,wewilltryit
andthenwe'lldotherealway.Butthis'llbeinterestingtoseehowcloseyouguys
get.
Yeahthat'sfine.Nowyou'regoingtomeasureeachoftheangles.Eachofthe
interiorangles.Ascloseasyoucan,doesn'thavetobeperfect.
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Allright,let'sstartmeasuringsomeangles.
Guys,getatleastoneofthosedownandmeasureitbeforeyoustartthesecond
one.Justsowecangetalittlebitofdatahere.
Youstartatthecorner,youlineitup.Sothisisyourbaseofyourangle,right?So
thenyourotherangleiscomingoffofthisside.Nowtheeasiestwayistouse
anotherstraightthingandthenlineitup.Nowbecausethislooksacute,sowhat?
You'regoingtousethetoppart?So78,sure.Nonono,that'scountingfromthis
way,thatwouldbe110.Socareful.
Youguysgotone?Oh,yeahthefouriseasy.Thentensided...Theyactuallywanted
ustogouptotwelve,which...I'mnotsurehowcrazythatwouldbe.Here'sthe
thingtodo,itdoesn'thavetobeallnice,likeyoucanmakethemgoinandout,you
canmakethemkindofdowhatever.Although,thosearekindofhardtomake.
Allright,what'dyoufind?Okay.541,ohcool.That's,itis540yeah.(laughs)That's
reallygoodthough.Wegotclose,allright.
Thesearegettingcloseguys.Let'stakeonemoreminute,getatleastonedown
andstartaddingupthoseangles.We'llcollectasmanyaswecanonourtable.And
thenwe'llfindtheexactones.
Youguysgetthetensided?Isthattoomany?Iguessthatworks.That'satricky
one.
Fivesidedone'sgood.
Yeah,justtoafivesidedoneforme.Let'strythatinstead.Yeahthat'sbetter.
Allright.Onceyougetoneofyouranswers,throwitinthetable.I'llgiveyouone
moreminute,we'llcollectwhatwe'vegot.Getatleastonedown.
Okaysolet'sstartmeasuringsomestuff,yeah?Actually,let'susetheedgeofthis.
Pickanangle.Onyourshapehere.Sure.That'sfine.Okayputyourprotractoronit.
It'sthewayyou'regoingtodoit,isyouseethelittleanchorrighthereinthe
center?That'slikethetarget,youputthatontheend,andthenyoulineupthis
littlelinewithoneofthesides.Somethinglikethat.Andthen,youwanttoseethis
line.Nowsincewecan'tseeitgoingalltheway,wewanttosortofmatchit.So
somethinglikethat,sowhatangleisthatgoingtobe?Thatwouldbeifyou're
comingfromthisside,butseehowit'sobtuse?Soit'sgottogotheotherside.Let's
goalltheway,so90,100.102soundsgreat.Allrightputitdown.
That'swhatwe'retryingtofindout.Whatdidyoursaddupto?Ithinkyou're
missingoneangle.Thisone,thisone.Youknowwhatmightbeeasier?Yeah,go
fromthisside,andthenyoujustminusitfrom360.Becauseofthefullcircle.That
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mightactuallywork.Sure.Sominusthatfrom360.Yeah,that'llwork.Yeahit'skind
ofhardtomeasuretheotherside.
Allright,dowehaveacoupleangleswecanputuphere?Iknowthegiantonesare
kindoftricky.Howaboutthetriangle?Wealreadyknowthatone,yeah?Student,
howaboutthetriangle?Whatisthatgoingtoaddupto?Rememberthatone?
180?Allrightgood.
Allright,Isawacouplepeopledothequadrilateral,thefoursidedone,although
we'vealreadytalkedaboutthat,sowhatdoesthatonecomeoutto?360.Allright,
nowIsawacouplepeopledoingthepentagon.Isiahyouhadthatone,whatdid
thataddupto?541,butyou'rethinkingitmaybe540.Allright,nowlet'sjust
surveyanyone,doesanyonehaveanothersumthatwecandorightnow?Anything
bigger.Iknowthesewerekindoftricky.Youguysaregoingtofinishup.Whatis
that?Sevensided?Eightsided.Okay.
Thereisapatternhere.Butit'skindofhardtofind.Whenyou'remeasuringthings,
you'renotgettingexactanglesbecauseyou'rejustmakingthisup.Thisisa
problem.However,whenpeoplehadlookedatthisthroughouthistory,they
noticedthattherewassomethinghappening.Therewassomesortofpattern.
Whentheytooktwodifferentpentagons,andtheymeasuretheangles,theywere
allaround541.Around540.539.Rightthere.Theywereallaroundthesamesortof
area.
Yougettheeightsidedone?880?Okay.Theywererelatedinsomeway.Youguys
canseeaswegethigheruphere,thesenumbersseemtogetbigger.Whichkindof
makessense,there'smoreangles.Thereissomesortofpattern.Yougetanother
one?Forwhichone?1020?Okay.Or1020.Okay.Youguyswanttoseethereal
answers?
Itisaddingbysomething.Youknowwhat,beforeIevenshowyouguysthis,let's
trysomethingelsehere.Let'strysomethingelsehere.Thisisinefficient,because
weliterallyhavetomeasureeverysingleangle.Butwealreadyknowafact,we
alreadyknowsomethingaboutpolygons,atleastsimplepolygons.Andthatis,
triangleshaveameasurement.SoifIdrawatriangleandIadduptheanglesofany
triangle,whatdoesitcomeoutto?180.Doesn'tmatterhowIdrawit.Doesn't
matterwhatit'sshape,it'ssides.No,right?Thisisafactthatwehaveproven,and
weprovethisseveraldifferentways.ThewayIrememberwastheparallellines,
drawthelittletriangleinthere.Buttherearelotsofwaysofprovingit.
Sohereiswhatwewanttodo.Let'sseeifwecantakethosepolygons,theones
youjustdrew,anddividethemupintotriangles.Nowhere'sthething,youcan't
justdividethemupintoanytriangles,becausethatwillnotwork.Youneedto
dividethemupnicely.
Letmeshowyouanexample.We'lltakethisguy.I'mgoingtodividethisupinto
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triangles,youguystellmeifIdidthisright.Notreally.Hereistheproblem.Iwant
thetrianglestohelpmefindtheinteriorangles.Thesetriangles,andthisisn'teven
atriangle...Thesetrianglesdonothavetheirverticesatthecornersofthisshape.
Whichmeanstheverticesarenotgoingtobeabletoequalthecornersofthis
shape.Iwanttodrawitnicely.AndI'mgoingtoshowyouguysatricksoyoucan
usethisonyourdiagrams.Thetrickis,pickonevertices.Highlightoneverticeson
theshapesthatyoujustdrew.Doesn'tmatter,anyone.Andthenconnectthat
verticestoeveryothervertices.Nowclearlytheadjacentonesarealready
connectedsonotnecessary,butforthese,connectit,andthenconnectit.Forthis
one,Iguessthere'sonlyoneconnected.SoI'mgoingtogiveyouguysaminute,
thiswillbemucheasierthantryingtoaddupallthisstuff,takeyourtwoshapes,
andconnectonecornertoalltheothercornersthatyoucan.Andthencountup
thenumberoftriangles.Giveitatry.
Dependinghowyoudrewit,itcancomeoutkindofweird.Justmakesureit
connectstothecornersandyoushouldbeokay.Yeah,thatworks.Sometimesit's
hardwhenyoudotheconcaveone.Butthat'sokay.Thisoneisweird.Thereyou
go,that'sagoodonetochoose.Good.
Allrightsowhatwe'redoing,we'reconnectingoneverticestoalltheothercorners.
Sothetrianglesarenotoverlappingeachother.Exactly.Nowthethingis,thisone
isalreadyconnectedwiththesideofthetriangle,orthesideoftheshape.Butyes,
theotheroneslookinggood.
Thatsoundsmuchmorereasonable.Verygood.Idon'tremembertheanswers,but
yeahyouwereactuallycloser.
Allright,let'sfilloutourtable.Dependingonwhichoneyouhad,wecanusethem
allandfillthisout.Allright,soturntopage150.Sothisisthemuchbetterwayof
investigatingthis,itdoesnotrequirethatyoumeasureanyanglesjustlikeinsidea
triangleitdoesnotrequirethatyoumeasurenayangles.Let'sfillthisguyout.Get
ourtheorem.Allright,so,numberofsides,verysimple.Doesitstopatoctagonor
didIcutthatoff?Okay,sowe'rejustgoingtoseethepattern,youcanextendthis
asfarasyouguyswant.Allright,numberoftrianglesformedbythediagonals.Fora
triangle,Iassumethat'sone.Howaboutforquadrilateral?Two.Howaboutfora
pentagon?Youguysseethepattern?Okay.Nowwhatisthesumoftheinterior
anglesofonetriangle?OkaysoI'mjustgoingtodothat.Nowthinkaboutthis,the
trianglesmakethepolygon.Thetrianglescornersmakethecornerofthepolygon.
SoifIhadupallofthosecorners,inallthedifferentsides,whichineffectisadding
upalloftheanglesinsidethetriangles,Ishouldgetthefullsumoftheinterior
angles.
Soforatriangleit'sobviously180.Howaboutforaquadrilateral?Howaboutfora
pentagon?Okay,keepgoing.Filloutthetable.Followthepattern.Mucheasier
thanmeasuringallthisstuff.
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Bytheway,youguyswillbeusingthistablequiteabitduringtodaysoyoudon't
havetocalculateiteverysingletime.Makesureit'sallfilledout.
Doyounothaveyourbook?Okay,don'twriteinthisone,butyoucanuse...What
arewedoing?Perfect,perfect,yeah.Butjustsoyoucanseeiteasier.Sobasically
thisandthisaretheimportantparts.
Allrighthexagon?720.Heptagon?900.Octagon?Nope.Idon'tknowwhy.That's
notevenatime,whatamIdoing?Allright,bytheway,let'sjusttakeaside...Do
youguysrecognizethesenumbers?Youeverseenthesebefore?Anyone
skateboard?Youguysknowthesenumbers?TonyHawk,superfamousfordoing
the900,yeah?Ithinkonsnowboardsandstufftheydo1080'sandmore.Stufflike
that.Thisissomethingthatissortoffamiliartomanyofyou,right?InXgamesorin
theOlympicsorwhatever.Whensomeonespinsonaskateboardorasnowboard
orskisorwhatever.Theydon'tusuallylandsideways.That'snotusuallyagoodway
toland.Sotheyusuallyareeitherlanding,forwardorbackwards,whichisaspinof
180.
Soifyoukeepadding180,aswe'vedonehere,yougetthosemultiples.What
comesafter1080?Whatcomesafter1080?1260?Sotheykeepgoing,and
obviouslyyoucanmakethisgoaslongasyouwant.Youcanfindanyofthem.But
that'skindoftedioustojustkindofadding180forever.IfIsaidallrightIhavethis,
100sidedpolygonhowarewegoingtofindtheangles?Eh,kindofalot,solet's
seeifwecanfigureoutabettermethodfordoingthis.Sorry,bytheway,here's
whatwejustdid.Uptothis.Youcanseethetriangles,youcanseethenumberof
sidesofthepolygon.Andthereissomemethodhere.Soifyouguyscouldanswer
thosequestionsonthebottomofthatpage,howmanytriangleswouldyouneed
fora22sidedfigure?Whatwouldtheanglesbe?Andthenseeifyoucouldcome
upwithasimpleformulaforthis.Andwe'lltalkaboutthatinasecond.
Whichonesarethese?FourthroughsevenIguess.Giveitatry.
Ifthat'stheangle...Orifthat'sthetrick.Seeifitworksforoneoftheseguys,but
yeah.Lookfortheformula.Whatdoyouthink?22sidedone.Howarewegoingto
getthat?Withoutlikemakingourtablegoforever.Thinkaboutthis,thinkabout
this.Lookatthisguy.Lookatthisguy.So22sided,whatistherelationship
betweenthesetwothings?Whataretheydoing?Sixandfour.Seven,five.Eight
andsix.Istherelikeaneasywayofgettingfromonenumbertotheother?What
canImultiplythisby?Ignorethe22,let'sfindthepatternfirst.Sowehavefrom
fivetothree.WhatdoIgottodotofivetoturnitintothethree?Okay,didthat
workforthenextguy?Okay,sowhataboutthe22?Easy.
What'sthepatternhere?WhatdoIdotothefivetogettothree?WhatamI
adding?WhatdoIdotothefivetoturnitintothethree?Ohminustwo.How
aboutthesixtogettothefour?Howitworks.Howaboutthetwentytwo?Get
them.
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Youfoundit?Sowhat'syourformula?Yep,yep.Sohowareyougoingtowrite
that?Okay,andthenwhatareyougonnadowiththattogettheanswer?Gotit.
Yep.Allrightyougotit?Oh,giantnumber,okay.Okay.
Allright,Iwanttoseesomegoodequationshere.
Allright,how'dyouget21?What's22-2?Dowehaveonemore?Sothat'sthe
question.Thatwasagoodthought,that'sthequestion.Let'slookatourpattern
here.SowhenIhadfoursides,whatdidwedotothefourtoturnitintoatwo?
Likewhatmathdidwedo?Ithinkyousaiditoriginally.Minustwo.Didyouthen
addanothertrianglebackon?No.Okay,sononotnecessarily,youwererightthe
firsttime.Cool.
Youguysgotit?WaituntilStudentgetsback.
Sohowmanytrianglesdidyouguysmake?
Allright.Let'stakealook.Howmanytriangles?Twenty.Asyouguysnotice,the
patternisminustwo.There'sareasonforthisbutwedon'thavetogointoittoday.
Sothesumoftheinteriorangles?3600,right?Youtakethat20andyoumultiplyit
byhe180.Allright,solet'swritedownourtheoremhere.Ofcourse,youwantto
putthisinyourbookafterwardssoyoucanuseitbutlet'sjustwriteitdownhere.
Sobasically,thetheoremforthesumoftheinterioranglesofanypolygon,thecall
itanNsidedpolygon,IsawmanyofyouusingX,totallyacceptable,buttheofficial
oneisN.Soitsays,thesumofthemeasuresoftheinterioranglesofapolygon
withNsides,is(N-2)*180.Thatgivesyouthenumberoftrianglesandthenyou
timesitbythe180,whichistheanglesinsidethattriangle.Allrightpleaseputit
down.Andthenlet'spractice.Sothere'syourformulafortheday,nowyoucansee
howwegotit.Thatiswhatwe'regoingtousefortherestofourlesson.
40...Okaywegottime.
Ishouldprobablyputthisontheboard.
Allright,let'sgiveitatry.I'vegotapolygonhere,somepolygon,let'sfindthe
missingangle.Workwithyourtablepartners,you'vegothirtyseconds.Let'sgo.
Youcanuseacalculatorofcourse.Let'sseewhatyougot.Sowhat'sitgoingtoadd
upto?Howmanysides?Checkyourtable.Let'sseewhatyougot.
Easy.Youminusfrom360?(laughs)Yeah,easy.
Allright.Twentyseconds.Yougotthem?Letmesee.What?It'sactuallylikeapen.
Itlooksdriedout,yeah.Itlookslikeit'sleakingandstuff,Idon'tknow.Okayif
you'vehadthatsinceseventhgradethenit'sdefinitelynotgoingtobeworking.
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(laughs)
Allright,what'dweget?64.Allright,sowhatkindofshapeisthis?Quadrilateral,
youcountthesides,thereforewecheckourtable,360degrees,addthoseallup,
minusthemfrom360.Here'sanothertrickthatIliketodoguys.SometimesIforget
thetable,sometimesIforgetthestuffsohere'swhatIliketodo.Iwilljustliterally
drawthetriangleonthepaper.SonowIcansee,ohyeah,twotriangles,180+180
right?AndI'mdone.Thatisamuchnicerwaythanhavingtoreferenceatableor
memorizesomething,justgenerateitforyourselfaswegoforward.
Allright,we'vegotonelastthingtotalkaboutbeforeIletyouguyspractice.We
havesomespecialpolygons,wehavesomepolygonsthatarenice.Thesearecalled
regularpolygons.Soaregularpolygonisapolygonthathasequalsides,equal
angles,weknowtheequilateraltriangle,weknowthesquare.Butofcourseallthe
otherpolygonshavethesameproperty.Theydon'treallyhavefancynames,we
justcallthemregularpolygonsbutyoucanseewehaveabunchofdifferenttypes.
They'reallequal,obviouslytheiranglesaregettingbiggeraswego.Andtheyhave
somespecialpropertieshere.Themainpropertyistheirangelsareequal,likewe
said.Whichmeans,wecanfindthemeasureofaspecificanglewithouthavingto
actuallymeasurealltheotherones.Thatworksoutprettygood.Allright.Let'sseeif
wecanfigurethisout.Igottwopolygonshere,workwithyourtablepartners,find
metheangleorthemeasureofasingleangleinthatpolygon.Remembertheyare
allcongruent.Soverysimilartohowyoudidtheequilateraltriangle,let'sseewhat
youguysgot.
Student,easyhuh?Perfect.Gotthem.Sowhatdiditcomeoutto?Good.Forthe
firstguy?Okay.Let'sfindBnow.allright,Iseepeoplehavegottheansweralready.
Useyourtables,references.
What'dyoufindJordan?Thatlooksawesome.Idon'tremembertheanswertothis
one,butthatlooksaboutright.How'dyoufindthisone?What'dyoudivideby8?
Perfect.Gotit.
Allright,Iseesomeanswersalready.Allright,Dustin,A?72?How'dyouget72?Oh
okay.Soyou'reontherighttrackthough.You'reontherighttrack.Sohehada
sum,andhedivideditbyacertainnumberofsides.Ithinkyoujusthadthewrong
angle.Solet'sfixit.Whatisthesumofthistypeofshape?Isit540?Sofiveside
shape,540.Andwedivideitby?5.Becauseit'sfivesides.They'reallequal,and
thatgivesus?108.Verygood.So108shouldbeforA.AndthenforB?135.So
whateverthisis,wasit1080?Thefullone.Somethinglikethat.Andthataddsupof
courseto1080andyoudivideitby8.Thisistheotherthingthatyouguyswillsee,
theywillnotgiveyouanyangles.Theywillgiveyouashape,andyouneedtodo
thedivisionyourselftofindthoseangles.Thesearekindofnicethough,right?
Nowinthebooktheyhavethistable...I'mtryingtothinkifwehavetimetodo
this.Ithinkwe'regoingtoskipthisfornow,youcangeneratethemasyougo.We
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mayhavetimeattheendtofillthisout,butIdon'twanttodoitrightnow.
Basicallytheywantyoutokindofgeneratethoseinteriorangles.Nowlet'sjustput
afewoftheminheresinceyouguysknowsomeofthem.Wesaidthe8onewas...
Whatwasit?135?Fortheeightsidedone,wejustsaid135.Andthenthepentagon
was?108?Howabutafoursidedshape?Youguysknowthisone.Whatdowecall
aregularquadrilateral?What'sthefancyname?They're90,sowhatdowecallthe
shape?Theyare90,butwhatdowecallthatshape?Ofcourse,asquare.Andwe
havethetriangle,youguysknowthatone.Thereyougo.Allright.Again,don't
worryabouttherestofthese.Youcangeneratethemasneeded,we'llusethem
later.Ifyouwanttofillthisoutwhenwe'redoingourpractice,that'sfine,Ijust
wanttomoveonforasecondthough.Aswegoforward,youcanfinditeachtime.
BasicallyyoujusthavetofindthatsumandthendivideitbyN,whateverthe
numberofsidesare.Orthenumberofanglesinthiscase.
Iwanttotalkaboutonelastthing,Isawsomeofyouguysdrawingthingsthatlook
likethis.Isthisapolygon?Yeah,that'sfine.That'sapolygon.Howisitdifferent
thanthenormalkindthatwedraw?Somethingaboutashape,yeah?Something
aboutitsanglesactually.Yeah,ithasthesereflexangles,right?Sothereflexangles
arebiggerthan180,andthey'reinteriorlikethatsothey'rekindofhardto
measure.Butthey'restillangles,andthisisstillapolygon.Thisisnotapolygon.
Whenyoudrawitsoitoverlapsthecorners,thatsuddenlybecomesseparate
polygons,right?Apolygonmusthaveacontinuousarea,whichmeansthatallthe
areainheremustbeconnectedtoitself.I'mgoingtoshowyouguysalittle
animationhere,justtoprovetoyouguysthatthisworked.WhatI'vealways
learned,orwhatthey'vealwayssaid...Iwaslookingatatextbook,theyalways
say,usethesepolygons.Right?Useaconvexpolygon,don'tusetheconcaveones.
AndI'mlike,wellwhynot?It'sstillapolygon.SoIfoundthislittleprogramthat's
goingtoanimatethis,andlet'ssee.Let'sseeifitworks.SoIwantyouguystolook
downhere.Thesearetheangles,thecomputerisjustmeasuringthem,andthen
it'saddingupthosethreenumbers.Solet'sseewhat'shappeninghere.
Okay,anglesarechanging.What'shappeningtothesum?Rocksolid.Youcanmake
itreallytiny,reallybig,butweknewthat.Triangles,obviously.Let'slookatanother
typeofshape.Howaboutthisguy?Youcanseethesum,540,likeweknow.Let's
messitup.Notchanging.Evenifwemakeitcrazylikethis.Rocksolid.Thisisthe
coolthing,doesn'tevenmatterwhattheshapelookslike,unless...DidIdoitin
thisone?Ican'tremember.(laughs)Thatworkstoo.Unlessyouoverlapit,likethis,
andnowsuddenlyitcan'tcalculatebecauseit'stwotriangles.Right?Itdoesn't
work.Youguyscansee.Andofcourse,foranyshape,this'llwork.Here'sthe1080,
theeightsidedshape.MakePacman.Anycrazyshapethatyouwant.Alltheangles
move,butyoucansee,asoneofthemgetsbiggeritsortofforcestheotheroneto
getsmaller.Right?Nomatterwhatcrazyshapeyoumake,itwillstillwork.Unless,
ofcourse,youcrossoneofthemtonotmakeapolygon.Idon'trememberifIdo
thatforthisone.Allright,youguyscansee.Eventhatonewouldwork.Theyusually
don'tgiveyouonesthatlooklikethis,butyouguysgettheidea.Itstillwouldwork
evenifyouhadit.
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We'lldotwomorethingshere.Guesswhoneedshelp.(laughs)Guesswhoneeds
helpmakingalittlebench.Yourfavorite.Grandma'sgotabench,gottogowiththe
gazebo,youguysremembertheshapeofthegazebo?Wasitahexagon?Ithinkit
wasahexagon.Soshewantsabenchtogointhemiddle,tokindofmatchthat
gazebo.Hereisthebench.Youhavetocutthepiecesofwoodtonailtogether
here,sowithyourtablepartners,Iwantyoutofindmethisangle.Whatangle
wouldIhavetocutthat,sothesetwothingswouldkindofjoinnicely?Youguys
cangetthisprettyquickactually,sinceyouknowalltheotherstuff.Allright,giveit
atry.
Whatisthisblacksortof...Thisanglerighthere?ThatIhavetocutdowntheside
there.Somaybedrawthehexagonfromthetopifyouneedto.Seewhatyouguys
got.
Idon'tactuallyrememberwhattheanswerisforthisone.Whatwasthe...Ohwe
neverdidthehexagon.
Hexagonshaped,seewhatthatangleisgoingtobe.
Okay.I'mgoingtopassoutsomethingwhileyouguysaredoingthis,justkeepiton
yourdesk.
What'sthat?Idon'tremember,maybe.Why?Wereyougoingtosay60?Okay.
Okay.Ohinteresting,thatwouldwork.Verygood.
GivethattoStudent.
How'severythingcoming?Arewegoingtofindthisguy?Weknowit'sahexagon...
Maybelet'sfindthis.Thismaybeastrategy.Whatifyoufoundthisangle?Thebig
angle.Seeifyoucangetthatguyfirst.
Allright,Natalie,yougotit?60.How'dyouget60?Okay.That'sthefullinterior
angles.Noyouwererightthefirsttime.Youtookthe720whichistheinterior
anglesofahexagon.Youdivideditbysix.Thatgivesyouthisangle.Thisoutside
angle.Butthat'snottheanglewe'recuttingwithoursaw.Sohow'dyougetthe
actualangle?Exactly.Andyouguysrememberwetalkedaboutthis,whenyou're
doingliketrimorwhateverinahouse...Wewatchedthatvideo.Youwanttomake
itequal,nomatterwhattheangleis,yougottobisectit.Sotheeasiestwayof
courseisjusttoknowtheangleanddivideit.Butrememberwehadthosepaper
tricks,orthecompasstricksandeverything?Sothatoneworkstoo.Allrightso,this
areallifeapplication,I'vehadtodostufflikethiswhenI'vebeenbuildinglittle
homeprojectsandstuff,andofcoursegrandmaappreciatesyourguyshelp.
Onelastpiece,whatifwehavetwopolygonsthatarestucktogether?Soletsdo
thisonetogether.Andthenwe'regoingtodosomegroupworkandpresentafew
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problemsattheendhere.Let'sfigureoutwhat'shappeninghere.Wegottwo
regularpolygons.Wehaveasquareandahexagon,wewanttofindthisangle.This
sortofoutsidecurvingangle.WhatshouldIlookforfirst?Theinteriorangles,sure.
Theotherside.Howaboutforthesquare?Wegot,wegot90degreesrighthere.
Howaboutforthehexagon?Wejustdidthisone.120.Sowhat'snext?What'rewe
goingtodo?How'dyougetthat?Soyouaddedup90+120.Andyougot210.So
that's,that'sthisangle.Andthenwhat?How'dyouget150?Oh,wrongcolor.So
youwent360-210,andyougot150.Allright.
Sothisistheotherskillandwetalkedaboutthislastclass,manytimesyoucanfind
partofacircleandthenyoucanusealittlesubtractiontofindtheothermissing
partofthecircle.justlikewhenwehadalinearpair.Youcanfindthatmissing
anglethataddsupto180.Theseanglesaddupto360.Andsoyoucansortof
generatethatbysubtractingitofftheend.
Thesearethebasicskillsthatweneedtobesuccessfulatthis,we'regoingtobe
solvingsomeproblems,sograbthepaperinfrontofyou.Iknowit'skindofsmall,
soifyouneedroompleaseuseyourbook,thisisgoingtobeourpractice.Let'stake
...Whatdidissay?Allright.We'lldoabout15or20minutes.We'llseehowlongit
takes.Idon'tknow.There'sacoupleproblemsonhere,therearesomeequations
sopleasebecareful.Youmaywanttowritethefullequation.Iknowyouguysare
goodatjustsortoffindingsomeofthisstuffinyourhead,butifyoucouldwrite
thefullequationoutinyourbook,thatmayhelpyoufortheoneswithalotofX's
inthem.Justbecarefulwiththat.I'llturntheradio,wehave15minuteslet'ssay.
Pleaseworkonyourhandouts,workquietlywithyourtablepartners,andbeready
topresentacoupleattheend.We'llthrowsomeupontheboard.
IsitallrightifIturntheradioonalittlebit?Theyliketoworkwithsomemusicso...
(musicplays)
Howyoudoing?Nowwouldactuallybeperfect,sojusttakethis,I'llwriteyoua
pass.
Justfinishthatupathome,thenthere'sahomeworkpagerightherethatyoucan
workon.
Allright.Howyoudoing?Sowhatareyouthinking?Whichonedoyouwanttostart
with?Sure.B.OhD.Excellent,okayoohinteresting.Basicparallellines,okay.Sowe
wanttofindX.Youseeyou'vegotparallellines,isXequaltoanyoftheanglesover
onthissideoftheparallels?Gottobe,they'reparallellines.Ignorethisthen,don't
worryaboutthepolygons.Justparallellines,whichangleisXgoingtobeequalto?
Whichone?Righthere?Thisside.Ohperfect,becauseit'scorresponding,okay.So
wegotthatguy.Anyothers?Thissidetoo,okay.Somaybethisisyourstrategy.
Maybeyougofromthatguyinstead.Allright,showStudent.
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Thatwouldbeyourstrategytofindthesum.Yes.Ifyouwantto.Thenyoudon't
needtodothetriangles.Youdefinitelydon'tneedtrianglesforthat.Here'sthe
problem.Ifyoustartputtingtrianglesinhere,yourchoppingupthatangleinto
somethingthatwedon'tknow.Intoliketwopieces,youknowwhattheirsumis,
butyoudon'tknowwhateachseparatepieceis,soyoudon'tneedthat.Whatdo
weknowaboutalloftheseangles?Whatyou'resayingisthat,thisandthis,equals
403.Rightthatdown.LikeIsaid,sometimes...Likethisone,youdon'tneedto.This
oneyouliterallyneedtowriteanequation.Yeah,whereveryouhaveroom.
Allright.Youdoingokay?Isit180?Letsthinkaboutit.Howmanytrianglesare
insidethisshape?Eachtriangleis180.Sowhataretheyalltogether?Youdon't
havetodoitinyourhead.You'vegotthreetriangles,they'reeach180.That'sa
greatwaytodoit.That'swhattheyalladdupto.Sonowcanyoufindthisangle?
Yeahlet'sdoit.
Howyoudoing?Excellent.Sothewholethingsaddsupto...Oh.Careful.Careful.
Rememberthethingwetalkedabout.Withinthisshape,wehavehowmany
triangles?Four.Right.Twolessthanthenumberofsides.Wehavefourtriangles.
That'swhyyouhavetodothe180*4.Otherwiseit'dbetoobig.Giveitatry.Ithink
Iactuallyhaveone,letmesee.Ahyes,hereyougo.Justgiveitbackattheend.
Sowhat'sthe403?That'swhattheseguyswere?Thosefour,minusfrom900,gives
youthat.Okay.Excellent.You'resayingthatthisguy,andthisguy,andthisguy
equalthat.Okay.Let'sfindtheN.Writethatequation.
HowyoudoingStudent?Ohyeah,gothem.Let'slookatthisguy.Thisguy'spretty
good.Allright,whatkindofshapeisthis?Heptagon.Sevensidedokay,sowhatdo
theyalladdupto?Okay.Startminusing.
Yougood?Whatkindofshapeisthis?Yeah.Useyourtable.Justallgofromthe
samevertexthough.
You'resaying,ignoringthisasatriangleandthisasaseparateshapelikethewhole
thing?Let'ssee.One,two,three,four.Sure.Here'stheonlyproblem,youknow
thisangle...Oractually,you'renotevencountingthat,you'recountingthatasa
wholeside.Sowhatisthisangle?Thisis90.Butwhatisthispartofit?You'regoing
tohavetoaddthatparttothe90.Didyoufindthat?Sokindoflikeourbellwork.
Thatwasagreatstrategy,doingitinonestep,butfindingthispartfirst,andthen
doingyourwholesubtraction.Yeah,thatwouldtotallywork.Dotheytriangles
separately,andthenputyournumbersinandstartdoingitfromthere.
Howyoudoing?Whichoneyoulookingat?Oheasy.Yeahthat'sactuallyagood
one,youcandothatoneforasecondone.Alotofregularpolygonsinthere,works
outgood.
Howyoudoing?Youcandothisonefirst,that'sfine.Cool.
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Howyoudoing?Whichoneyouon?Youcanjumpdownthere,that'sfine.Don't
forgetsomeofthese.Iknowthere'ssomeequationsinthere,neverfun.What's
yourstrategy?Ohyoualreadyfoundthem.Ohperfect.Andyoudid...Youadded
them.Yeah,that'sgood.
Let'sseeyoudid...Wait,how'dyouget360?Canwetrythatagain?What's180*4?
Sowehavea...What'sthis?Asixsidedshape?Soit'sahexagon?Justtryiton
here,what's180*4?Justtryit.Seewhatyougot.Thereyougo.Youcanseethat
wasthispartonyourtable.Youcandoit...Likeyoushowed.Youcandrawalittle
pictures,drawlittletriangles,andthenjusttimesthem.Okaynowthat'sthething
you'regoingtodivide.
Whichone'stricky?Thisguy.Okaysowhatkindofshapeisthis?Countitagain.
Howmanysidesdoesthathave?Howmanyisit?Seven.Okaysothatwouldbe?
Heptagon,yeah?Whatdoesthataddupto?900.That'swhereyoucanstart.You
knoweverythingaddsupto900,soyoucanwritethisnicebiglongequation,that
hasalloftheseaddingupto900.Youprobablywanttodothatinyourbooksince
there'snotatonofroom.
Whatdidyoufind?Ilikehowyoudivideditupthough,excellent.Okay.Whatkind
oftriangleisthisguy?Whatkindoftriangle?Itisisosceles.Justlikethebellwork.
Here'syourfirststrategy?Findtheseguys.Wellallthreeoftheseequalto180.So
let'sfigureoutwhatoneofthemisgoingtobe.YouseethatStudent?Let'sfigure
outwhatbothofthemaretogether.Whatdoyoudowith...Tofindthat?Close.
Notadditby180.Thereyougo.Tryit,tellmewhatyougot.
Allright.Onesec.Let'stakefourteenmoreminutes.Helpeachotherout.We'll
presentacoupleofthese,atthatpoint.
Letmesee.Bothofthesetogetherare160.Howbigisoneofthem?Because
they'reboththesameremember.WhatdoIdoto160?Thesebothequal160.
Togethertheyequal160,andthey'rebothequaltoeachother.WhatdoIdoto160
tofindoneofthose?Thereyougo.Whatdoyoudivideitby?Howmanyanglesare
there?Theoneswe'retalkingabout.Yeah,whatareyougoingtodivideitby?Now
thatyouhavethat,youshouldbeabletousethewholeshape.
Allright,youguyshadaquestion.Presentit?Whichone'sthat?Sure,ifyoucan
showthefullequation.That'sfine.Okay.Giveitlike10minutesandI'llhaveyou
guyswriteit.Whichone?Idon'tknowdoyou?Ifyoudon'tneedit,don'tuseit.
Sure,whateveryouneed.Andthenyougotthat90.Butthenwehavetwomissing
pieces.Shoot.Webasicallywanttofindthisangle.Idon'tthinkit'sequaltoK,
necessarily.We'reatthispointwhereweneedtofindthisangle.Maybeweshould
usethetriangle.Thisis180,somaybeweneedtofindthisanglethen.Interesting.
Gotthem.
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Smart,mart.Ilikeit,Ilikeit.
Whichnumbersdoyousubtract?Theseguys?Butjustthesefour,yeah?Nowyou
knowwhatever'sleft...Wasitfour-something?Nowyouknow403isequalto,this
guyplusthisguyplusthisguy.Sowriteitdown.That'swhenyouhavetoactually
writetheequation.Youcan'tjustdoitinyourhead.
You'redone.You'redrawingsoyou'redone.Whichone'stricky?Firstone.Basically
whatyouhavehere,youhave403.Whatis403?Whatisthatequalto?Theangles
withNonthem.Let'swritethatdown.Here,doitinyourbooksoyouhaveroom.
Let'sliterallywritethatequation.Rememberyoucandosomeoftheseinyour
head,likethesecondone,butsomeofthemyouliterally...Youneedtheequation.
Whatarewegoingtowrite?403equals.Soit'sthesumofthatstuff,yeah?Let's
justwritethatdown.Putabunchofplusesinbetween.Seewhatitcomesoutto.
Okay,looksgood.Sowhatdowedowiththesetypesofequations?Youpretendin
NisXbut...Whichthingscouldyouputtogether?Yeah,let'sdothatfirst.Andthen
dealwiththenumbers.
Makesense?Good,good.
Pleaserememberguys,youcanalwayscheckyouranswerattheend.Onceyouget
anangelmakesure,isitobtuse?Isitacute?Doesitmakesensewiththetypeof
polygonthatyouweregiven?Seewhatyou'vegot.
You'regoingtousethisshape.Nowthatyouhave58,howbigisthisangle?Nono
no,it'sastraightline.Thissideis58,what'sleftover?Whatdoesalineaddupto?
Justastraightline.Whatistheangleofastraightline.Thereyougo.Minusit.Find
it.Thenyoucanuseyourlittletrianglething.
Isthatnecessary?Youcan,ofcourse,butdoesitneedit?It'seasierbecauseit
madeatriangle?Yeah,sure.Ofcourse.Youjustwon'tknowthattopangle.Doyou
knowthisangle?Doyouknowthisangle?Keepgoing.WellIdon'tknow,isit120?I
don'tthinkit's120.Whatkindofshapeisthebigguy?
Whatkindofshapeisthat?Countitup.Sevensides,youknowwhatweactually
callthatguy?That'stheoneIalwaysforget.Wegotaheptagon,whatdotheyall
addupto?900,okay.Canyouwriteanequation?Maybethathasequals900init
somewhere.It'sgoingtobereallylong,butit'llhavesomepiecesinit.Yep,put
themalltogether.Ofcourse,definentlytheN'sbecauseweneedthatinour
equationtosolveityeah?It'sgoingtobelong.Ifyouwant,Iknowsomepeople
addedthenumbersupfirst,liketheseparatenumbers,andthenputtheN'sjustto
makeitabitshorter.Sothat'salittleshortcutyoucando.
Sevenminutesguys.
Whichone'strickynow?E?Ilikethatyoulabelledthistriangle,youhavetheequal
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angleshere.ThatisdefinitelythefirststepIwoulddo.Sortoflikethebellwork.
Seeifyoucanfindthatmissingangle.Andthenyoucansortofstartdealingwith
thisentireshape.Ignoringthetriangle,onceyougetthislittleangleinhere.Look
backtoyourbellworkandseeifyoucanfigureoutwhateachofthoseanglesare.
Youfoundthem?Youfound,whatwasthisanglerighthere?Youhad58overhere,
youseethisoneStudent?Youdoingthesameone?Thisis58,nowwe'retryingto
findthisbigangle.Ithinkyoucanjustleaveit.Idon'tthinkyouneedtodivideitby
two.Ithinkwecanusethewholething.Yeahjustleaveit.Sonowhere'swhatyou
have.Youhave90degrees,youhave122,youhave88andyouhave96.Nowlet's
findK.Whatdoalloftheseaddupto.No,there'sthreetriangles.What'sitgoingto
addupto?Onesec.Youcanlookatyourlittletablehereifyouwantto.Three
triangles.Thereyougo.Startminusing.
Yougotsomeparallellines.That'strue.Whydon'twestartlabelingsomeofthose
unknownsaslikeXandYandstuff.Sorry,XandZthenIguess.Now,arethereany
otheranglesinthisdiagramthatareequaltoXorZ?Itis,verygood,let'slabelitZ.
Sure.Wesawtheproblem,becausewehavetoomanyangles.Sowehavetoseeif
wecanfigureoutsomeoftheseangles.Maybewecanwritesomethingalittlebit
differenthere.Yeah,somesortofequation.Thereisdefinitely180inthere,yeah.
Wedon't.CanwewriteXasaZ?LikeusingtheletterZinsteadofX.WhatisXasit
relatestoZ?Sure.Idon'tthinkyou'regoingtodothat.It'sthesame.
Ihavealotofwhiteout.(laughter)Iguessthat'sgood,butno.WhatIwasthinking
issomething...Andagain,Idon'tknowtheanswerbutsomethingintermsof...
YouknowthisisZ,andyouknowthisisrelatedtoZ.It'snotjustanothervariable.
Thisisthesame.Let'ssayZwas80,whatwouldXbe?IfZwas80,whatisX?How'd
youfindthat?Soyouminusitfrom180.That'sthestrategythatI'mthinking.I'm
justmakingupsomething.Itkindoflookslikeitcouldbe80.Butbasicallyyoudon't
havetowriteathirdvariableyoucanjustwrite180-Z.Thatmayhelpyou.Thatmay
helpyou.I'dgiveitatry.Idon'tknowwhattheactualansweris.Youguystellme.
Allrightwehave15minutes.Youwantto,youwanttostartwritingsomething?
Whatarewedoing?Whichone?Wait.Whichoneisthis?Thisone.Whichoneare
youdoing?Youcanwritesmaller...Wouldthishelp?Here,trywriting.Ijustwant
tosee,wecanchangeit.Doesthathelp?Isthattiny?Youcanzoomin,ifyouwant
tomakeitveryprecise.Andthenwecanzoominfortheguess.Andyoucan
changethecolorstooasyougo.
Allright,let'stakeaboutfourmoreminutes.Lookinggoodguys.Ifyoufinishearly,
feelfreetostartyourhomework.We'llpresentafewoftheseandtalkaboutthem
attheend.
No,homeworkis152.But,startonthisonefirst.
Allright,whatdoyougot?Idon'tactuallyknowhowtodothisoneeither.I'mtrying
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tofigureoutwhatthetrickiswiththis.Thesearen'tparallelarethey?NoIguess
not.Sohere'stheonlythingIcanthinkof.What'stherelationshipbetweenthose
twoangles?Gottheparallellines.Gotsomeangles.Idon'tknowmaybe.Trythem.
Imeandon'tjustchangeit,butseeifit'sequivalentinsomeway.Sowhatdowe
callthesetypesofanglesthatwehave?Kindoflookatitlikethis.Wehaveparallel
lines,we'vegotthetranversal,whatdowecallthoselines?Close.Interior,there
yougo.Sothey'reequalright.That'sthestrategyI'mthinkingwe'vegottogofor.
Allrightwhichone'stricky?Soyouknowwhatitaddsupto?900.Andyouminused
theanglestheygaveus.Basically,whatyou'resayingisthisisequaltothethree
remainingangles.Let'smakeanequation,theyadduptothis.It'sgoingtohavea
bunchofN'sinit,butthat'sfine,youcanputthemtogetherandseewhathappens.
Gotthesamething?Cool,letmeknowifyouhavequestions.
Letmeseewhatyougot.Whichone'stricky?Whichone?Whichone?Thisguy.No
thisguy,comeonwecandothis.Don'tbeafraidoftheN's.notevenneeded.How
manytrianglesfitinsidehere?Yeahstartwiththat.Allright,howmany?Orlookat
thetable.Sowhataretheangles?Addthemup.
There'sfivetriangles,eachofthemis180.Sowhatisthetotalsumgoingtobe?
Nope.Fivetriangles.Okayso900.Yeahthat'sfine,thankyou.So,900.Weknow
thealladdupto900.Soyoucanwritethisnicelongequation,alloftheseadding
upto900.
Thatisreallyzoomedin.Ohactually,whatyou...Basicallyifyouhitthisbutton,it
willlockittotheview.Soyou'llwanttoputitontheviewthatyouwant,andthe
hitthatbutton.It'lllockit.Wecanzoominlater.Don'tevenworryaboutit.
Probablyiswrong.Idon'tactuallyknowhowtosolvethisone.Weneedmore
information.Itcouldbe.Letmeseewhatotherpeoplegot,Idon'tevenknowthe
answertothisone.Itseemsreasonable,butthenthatmightbetoobig.Wellthat's
obtuse,sothatwouldworktoo.45*3.I'mgoingtosee...Letmeseewhatother
peoplegot.
YougotC?Let'ssee.Iknow,StudentdidyougetC?Idon'tknowifthisone'seven
possible.Ifeellikethere'snotenoughinformation.Whatwouldmakethiswork?
Didtheyforget...Ifthesewereparallelwoulditwork?Liketheothertwo?Yeah.
Okay.Idon'tknowiftheymeantthat,butlet'sputthat.
Canyou...canIdosomethingrealquick?I'mgoingtotalkwithyou,justmake
thesetwolinesparallel,putthelittlemarksonit.Okay.
Guyswe'vegottenminutes.OnequickthingIwanttoadd.Idon'tknowIthisis
howit'ssupposedtobe,Idon'tknowtheanswer,but...Canyouguyszoominon
C?IfyouguyslookatC,Idon'tknowifthere'senoughinformationtosolvethis
one.ThereshouldbebutIfeellikeI'mmissingsomething.Sojusttomakethis
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work,thewaythatwe'vetalkedaboutit...Ifyoucanmakethesetwolinesparallel,
thenyoucansolveit.Thenyouhavemoreequivalenciesthatyoucanthenwrite.If
youhavethesethenofcoursethisangleiscongruenttothisangle.Sothiswould
be180-3Y.Andsuddenlyyouhaveenoughvariablestosolveforthis.Sothat'swhat
Iwouldsayjusttomakethisalittlemoredoable.I'mnotexactlysurewhatthey
wantedustodoforthisonebut,theotheronescomeoutprettynice.Feellike
we'remissingsomethingforthatonethough.ProbablymymistakewhenImade
thecopiesbutIdon'tknow,we'llsee.
Let'sgiveitonemoreminute,finishupyourlastproblem,we'regoingtopresenta
fewoftheseontheboard,we'regoingtotalkaboutthem.Andthenwewilldoour
exitpass.Doinggoodguys.
Ifyoucouldputyourcompasses...Oryour,protractorsbackonthestickynotefor
thenextperiod,thatwouldbeawesome.Ithinkthereweretwooneachone.
Somethinglikethat,ifyoucandothatbytheend,that'dbeawesome.Thankyou
guys.
Youcanzoominonthat.Dowewanttodoonemore?Youwanttodoanother
one?Allright,yougotit,that'sfine.Writeitdown.Willyoufinishthatone?Yeah,
well,let'sdothattogetherthough.Youguyscanwriteitlivewhenwe'retalking
aboutit.Yeahwe'lldothatattheend.Whenwe'representingjustzoominsowe
canseeexactlywhat'shappening.
Yougotthem?What'ditcomeoutto?Ohwegotanequation.Soyou...Where's
theequalsigninthisequation?It'sanequation...Whatdidwesayallofthese
addedupto?Exactly.Thewholethingequals900,thenyoustartsubtractingstuff
offtogetthosenumbersdown.
AllrightwhichoneareyouonStudent?Samething.Lookatwhatequationhe's
making.Allright,giveyoualittlemoretime.
Allright.Wedoingbetter?Makingsense?Whichone'stricky?Thisone?Hurryand
finishthatone.Okay.
Allrightguys.Let'stalkaboutit,eyesupfront.CanyouzoomintoBforusright
now?Wait,didn'twedothatone?Wasn'tthatoneofmyexamplesonthenotes?
Talkusthroughit,how'dwefindX?Sosixsidedshape,what'sthefancynamefor
that?Hexagon.Theyhavesixsides,soyoufoundthe720whichwasthesumand
thenyoudividedbysix.Howdidyouknowthatyoucoulddividebysix?Wecan't
dothatforexample...Here,zoomoutofit.What'sthat?Thetheoremhadtodo
witharegularhexagon.Howdoweknowthisisaregularhexagon?Theanglesare
allX.Nowthisdoesn'tactuallyhavetobearegularhexagon.IcouldifIreally
wantedto,stretchthismiddleside...Moveitdown.Icouldstretchthismiddleside
outtobetwiceaslongandtheangleswouldnotchange.Sothisisnotnecessarilya
regularhexagon,buttheregularnessthatcounts,theanglesums,ortheangle
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equivalenciesthosearethere.Sowedon'treallyneedtoworryaboutthesides.It
doesn'thavethesideslabeledsoit'snotnecessarilyaregularhexagon.
ShowusF.Canyouzoominonthat?Allrightlet'stakealookatFguys,thisismore
ofthehomeworkquestionsthatyouguyswillsee.Basicallythereareabunchof
variablesinsidethispolygon.Solet'sseeifwecanfigureoutwhatishappening
here.Sofirstofall,whattypeofpolygonisthis?Let'scountthemup.Isthatalsoa
hexagon?Sixsides,okaysohexagon.Niceeasyone.Isthisaregularhexagon?I
assumenot.Nothingislabeled,so.Whatdidyoudo?Sothatisthesumofthe
interiorofahexagonlikewesaid.Excellent.SoXis14,butthatisnotanyofthe
angles.Sohowdidyoufindtheactualanglesafterthat?Verygood.Andyougotall
thoseangles.Didyoucheck,didtheyalladdupto720?Thatissomethingyou
coulddo,again,Itrustyourwork.Butthatissomethingyouguyscoulddotofind
thetotalsumtoseeifyouransweractuallymakessense.
NowacouplethingsthatIpointoutherethatIreallylike,Ireallylikethatyouhave
thefullequation,notjustpiecesofit.Youcankindofkeepyourselforganized.I
reallylikethatyoukepteverythinginparenthesis,soyoudidn'thavetoworry
aboutnegativesignsorstuffspillingout,andthenyouguyscansee...Acouple
stepswereskippedherebutthat'sfine,combingtheliketerms.Youwanttogetall
oftheX'stogethersoyoucandivideoffthatcoefficientandofcoursewemovethe
8first,divideit,andthenitcomesoutnice.Iguessitdoesn'thavetocomeout
nice,butyouknow,it'samathprobleminthetextbooksoIassumeit'sgoingtobe
okay.
Allrightawesomeguys,didwehaveonemoreorarewegood?Actuallyweare
almostoutoftime.
Solet'sjustgothroughthebottomhere.Thesearetheanswers,solet'sseewhat
wegot.Canwezoom...Therewego.Youguyscanseemostoftheseanglessuch
asB,A,C,E.Thesearetheinteriorangleswe'vealreadycalculated.Sothoseare
trivial.Someoftheotheronesthoughyouhavetodoalittlebitmoremath.For
example,I,we'vegotthissixtydegreeanglerighthereinthisequilateraltriangle.
AndI,issortofthatoutsideanglewhichissortofinbetweentheoctagonangle
andthetriangleangle.Soyoukindofjustdoalittlesubtraction.ThisisatrickI
wantyouguystobethinkingabout,andwementionedthislasttime.Manytimes
whenyouarefindingamissingangle,theywillnotlabelthethingyouactuallyneed
tofind.TheywantyoutofindI,buttheydon'ttellyou,ohyeahbutyouhaveto
findXoverherefirst.You'vegottofindthebigangleYsecond.Theydon'ttellyou
that,youhavetosortoffindwhatyouwant.Andthisisabalancingactbecause
youcouldfindfiftydifferentangles.Andforty-nineofthemwouldn'tbeusefulto
you.Youcouldfindthisangle,andthisangle,andthisangle,andthisangle,but
they'renotreallyuseful.Soyouwanttoconcentrateitaroundtheareathatyou
actuallyhavehere.
Awesomeguys.Grabthishandout,youcanholditfornow,finishitupifyouhave
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not.Grabyournotes.Let'sdoourexitpass.Actuallyifyoucouldcloseyourwork
book,Iwanttoseeifyouguyscandothiswithoutlookingatthetable.Justmake
sureyouputyourcalculatorsbackattheend.Ifanyoneneedstolookattheirtest,
comeseemeafterclass.Allrightready?Go.Showmebeforetheend.
Okayperfecttimeyouguys.That'swhatcalculatorsarefor,comeon.
Whatisthisshapecalled?Nonagon?Idon'tknow.Countthemup.Justcountthe
numberofsides.
Sohowmanysides?Howmanysides?Sohowmanytriangles?Inside?Remember
thetrick?Ifthere'sninesidesthere's?Howmanytriangles?Remembertheminus
twothatwedid?Soifthere'snine?Seventrianglessofindtheangles.Letmesee.
Thatseemsreallybig.Good.Good.Good.Good.Good.Good.
Allright,howmanysides?Nine.Howmanytriangles?Seven.Howmanyinterior
anglestotal?Sorry,thesumoftheinterioranglestotal?1260?Andthesumorthe
division?140.Verygood.Sothisisinterioranglesofpolygons.Weknowhowto
findthiswithoutatable.Wecandivideupourshapeintotrianglesandthenaddup
thesumofthosetriangles.Sobasedonsomethingyouguysalreadyknow.
Allright,thankyouverymuchguys,putyourprotractorsonyourcard.Calculators,
upfront.Haveawonderfulweekend.Comeseemeifyouwanttoseeyourtest.If
youwereabsentlasttime,youneedtotakethetestso.Ithinkthey'restillabsent.
Studentdidyouhavetotakeatest?
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