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10thMathSumAngles
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Allrightgiveyouacoupleminutes,findthosemissingangles.Grabacalculatorifyou
needit.Youguystakeanotherminute.[inaudible00:02:12]Yeah,buthowdoyouknow
that?[inaudible00:02:26]Alittlemore,alittlemore.Youdon'thavetodrawthe
[inaudible00:02:44]
Ineedtostudy.
Allright,yougottosubtractthis.[inaudible00:02:55][crosstalk00:03:27]Dothesetwo
angleshave[inaudible00:03:40]
No.
Thesetwoangles,solet'sthinkaboutallthe[inaudible00:03:45].Insteadoftryingto
measureeveryangle,whatdegreeisthis?
[inaudible00:03:59]
Allright,solet'strytofigureoutwhat[inaudible00:04:02].I'llgiveyouguysalittlebit
moretime,seeingsomegoodanswershere,somegooddiagrams.[inaudible00:04:26]
Let'stalkaboutit.Thisissomethingwe'vedonequiteawhileago.Weprovedthis
quartertwomaybe,interioranglesofatriangle.Today,andIknowitdoesn'tsayiton
thetitle,we'regoingtobeextendingthisintopolygons.Whathappensifwehavea
four-sidedshapeorafive-sideshapeoraten-sidedshapeorathousand-sidedshape?
We'regoingtoextrapolatefromthereandfigureoutwhataretheinterioranglesgoing
toaddupto.Thenofcourse,justlikeyouguysdidforthese,findthosemissingpieces.
Forthefirstguy,whatkindoftriangleisthis?
Isosceles.
Isosceles,ofcourse.Whichmeansthesetwoanglesare?
Thesame.
Congruent,yeahthesame.Wecanlabelthisx=degreesifwereallywantedto.Iknow
manyofyoudon'tneedtowritethefullequation,buttobepreciselet'swriteitout.We
knowthesumoftheinterioranglesisalwaysthesame,whichis180.Wetechnicallycan
writethisfullequation.Whetheryouactuallydidthisornot,thisisthethoughtthatyou
guyshadwhenyouweredoingthis.Basicallywehave34+X+X=that180.Ofcourse,what
youguysactuallydidisyouminused34andthenyoudividedbytwo.Intheend,when
wedidthat,whatdidwecomeoutwith?
73.
73degrees,good.Becausethisisanequation,oneofthegreatfeaturesthatwecan
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exploitiswecancheckouranswer.Ifyouhave73,orsomethingelse,whatifyouhad72
oryougotadifferentanswer?Youcanaddthemup.Youcanaddup73and73and34
andchecktoseeifthatis180.Nowagain,thisoneisfairlystraightforward,soprobably
notnecessary,butthatisawayofchecking,suchasonatest,justtomakesurethatyou
gotitright.
Secondone,righttriangle,sameidea.Wehaveour64+90+something=180,andStudent,
whatdidthatactuallycomeoutto?
26degrees.
26degrees,soundsaboutright.Fairlystraightforward,it'sbasicarithmetic.Sometimes
youhavetobecleverbecausecertainanglesarecongruentandtheydon'tgiveyouone
ofthoseangles,becausethatwouldjustgiveyoutheanswer,soyourequationneedsto
beabitmorecomplicated,butingeneralit'sadditionandsubtraction.Wewillgetonto
someofthoseequations,lateron,whereit'slike64+Xor2X-3,andyouhavetomake
thosebiggerequations.Wetalkedaboutthatinourreviewlastclass,butforthemost
part,alotofthemathyou'regoingtodotodayissimplyadditionandsubtraction.Nota
bigdeal,shouldbeprettygood.
Nohomeworklasttime,huh?Ihavenotgradedyourtestyet.Iwilldothoseassoonas
possible.Ifyoureallywanttogetideaofwhatyouhave,comeseemeafterclasstoday.
WecangothroughitandIdon'tthinkIcangradeitthatquickly,butwecantalkaboutit
andseewhatyouguysactuallygot.I'lltrytodothatthisweekend,assoonaspossible.
Todaywe'regoingtobeinvestigatingandthenwritingdownourlasttheorem,which
hastodowiththeinterioranglesofanyshape,notjustatriangle,butwhatabouta
four-sided,five-sided,six-sidedshape?We'regoingtostartwithalittleactivity,sograb
yourbooks.Opentoablankpage,whichIguessisthenextpage.Graboneofthe
protractorsthatIputonyourdesk.What'syourquestion?
CanIgetmybook?
Yes,goaheadit'sfine.We'regoingtoinvestigate,andifyouactuallylookinsideyour
workbook,onpage149,there'salittletable.We'regoingtofillthatoutlater,somaybe
youjustwanttokeepthatofftotheside.Itlookslikethis,bytheway.Canwedouble
check?Isit159?
150.
Thereisoneon150,butIthinkthere'sanotheroneon149.Check149.Isthattheone?
No,thereyougo.Okay,yes.There'salotoftableshere.We'renotgoingtofillthemall
out.There'sbasicallytwoideaswe'regoingtodo.Firstofall,we'regoingtoinvestigate,
andthisistheinductivereasoning.We'regoingtophysicallydrawsomeofthese
polygons.We'regoingtomeasuretheirangles,whichagain,isnotthemostaccurate
wayofdoingit,becausewe'rejustgoingtomakethemup,andwearegoingtoaddup
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thoseangles.We'regoingtotrytocomeupwithanidea,comeupwithatheoryabout
howtheseanglesarerelatedinsideabunchofdifferentpolygons.
Thebookwantedyoutodoallofthem,andobviouslythat'sgoingtotakeforever,soI
justbrokeyouguysupbytwopolygons.Ialternatedthemsoyou'renotdoingamillion
differentsides.Onyourtablethereshouldbeastickynote.It'llsaypolygonwith9and5
sides,or7and4sides,whatever,justtwodifferentpolygons.Whatyouneedtodois
justlikeIdidoverhere.Makeapolygonwiththatnumberofsides,andthenmakethe
secondpolygonwiththatnumberofsides.Justmakethemstraight.Usethestraight
edge.Theydon'thavetobeperfect.Theydon'thavetoberegular.Theydon'thaveto
haveanyparticularlength.
Justmakethemandthen,asaccuratelyasyoucan,butroundedtothenearestwhole
number,pleasemeasurethoseanglesandaddthemup.We'lltakeacoupleminutes
hereandseewhatyouguyscando.Youcanworkwithapersonatyourtable,butdraw
them,measurethem,andthencompare.Seeifyouguysgotthesamemeasurements
foreachoftheseitems.Takeacoupleminutes.They'renotgoingtocomeoutperfect,
butwe'llgetthemcloseenough.[crosstalk00:10:58]
Yeah?Youhaveyoursixside,nowyouneedtomeasure[inaudible00:11:49].You're
doinggood.Youlineitupasmuchaspossibleandthenyouwanttoseewherethis
ends.[inaudible00:12:06]Youhavetostartfromthisside.
Pleasebecarefulguys,theprotractorhas2sidestoit.Right?Oneofthemcountingup,
oneofthemcountingdown,lookattheangleandseeifit'sacuterorobtuseandthat
willbeaneasywaytoswitchtotellwhichsideyoushouldbeusing,right?Justbe
carefulthat.Iswitchthatupallthetime.[crosstalk00:12:40]
Letmegiveyouoneofthese.It'skindofhardtomeasure[crosstalk00:12:46].Basically,
youaddthisandlineitupwiththesidetoseeiftheothersideaddsup.Ascloseasyou
can,I'mnotsurehowaccuratethisisgoingtobe.Wewilltryit,thenwewilldothe
rhythm.Thiswillbeinterestingtoseehowcloseyouguyswillget.[inaudible00:13:35]
Ascloseasyoucanget.Getatleastoneofthosedownandmeasureitbeforeyoustart
thesecondonejustsowecangetalittlebitaheadhere.
Youstartatthecornerandlineitup.Thisis[inaudible00:14:06]Right?Thenyourother
angleiscomingoffthisside.Theeasiestwayistouseanotherobtusetostraightenit
andthenlineitup.No,nothat'snotobtuse.[inaudible00:14:46]Yeah,thetenthskind
of[inaudible00:14:51].Itdoesn'thavetobethat.It'sgettingcloseguys.Let'stakeone
moreminuteandgetatleast1downandstartaddingupthoseangles.Let'sdoasmany
aswecanonourtableandthenwe'llfindtheexactangle.
Onceyougetoneofyouranswers,startthetable.I'llgiveyou1moreminuteandthen
we'llbedone.Getatleast1down.[inaudible00:16:22]Whatdoyoudo?[inaudible
00:16:36]Putthatontheendandthenyoulineupthislinewith[inaudible00:16:41]
Thenyouwantthisline,youcannotseeitalltheway.Thatwouldneedtocomefrom
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thisside.[crosstalk00:17:14]Allright,dowehaveacoupleangleswecanputuphere?I
knowthegiantonesarekindoftricky.
Howaboutthetriangle?Wealreadyknowthatone,yeah?Howaboutthetriangle?
Whatdoesthataddupto?Isawacouplepeopledothequadrilateral,the4sidedone,
althoughwe'vealreadytalkedaboutthatandsowhatdoesthatonecomeoutto?360.
Alright,Isawacouplepeopledoingthepentagon.Studentyouhadthatone,whatdid
thataddupto?
541,butI'mthinkingitmaybe540.
541,butyouarethinkingitmaybe540.Let'sjustsurveyeveryone.Doesanyonehave
anothersumyoucandorightnow?Anythingbigger?Student,youcananswer.Which
onewasthat?77?87?
87.
87?Okay.Thereisapatternhere.It'skindofhardtofind,allright?Whenyou're
measuringthings,you'renotgettingexactangles,becauseyou'rejustmakingthisup.
Thisisaproblem.However,whenpeoplehadlookedatthisthroughouthistorythey
noticedthattherewassomethinghappening.Therewassomesortofpattern.When
theytook2differentpentagons,andtheymeasuredtheangles,theywereallaround
541,540,539.Theywereallaroundthesamesortofarea.Wereyougoingtosay
something?
Igot880.
880?Okay.Theywererelatedinsomeway.Youguyscanseethataswegethigherup
here,thesenumberseemtogetbigger,whichkindofmakessensethere'smoreangles.
Thereissomesortofpattern.[inaudible00:19:35]
Igot1280.
Forwhichone?
Forthis87.
1020?Youguyswanttoseetherealanswers?
Isitaddingby1?
Itisaddingbysomething.Itisadding.Let's...BeforeIevenshowyouguysthis,let'stry
somethingelse.Thisisinefficientbecausewehavetoliterallymeasureeverysingle
length.Wealreadyknowafact.Wealreadyknowsomethingaboutpolygonsatleast
simplepolygons.Thatis...Triangleshaveameasure.IfIdrawatriangleandIaddupthe
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anglesofanytriangle?Whatdoesitcomeoutto?
180.
180.DoesitmatterhowIdrawit?Doesitmatterit'sshape,it'ssize?Noright?Thisisa
factthatwehaveproven.We'veproveditseveraldifferentways.ThewayIrememberis
theparallellines,youcandrawalittletriangleinsideit.Therearelotsofwaysofproving
it.Hereiswhatwewanttodo,let'sseeifwecantakethosepolygons,theonesyoujust
drew,anddividethemupintotriangles.Here'sthething,youcannotjustdividethem
upintoanytriangles,thatwillnotwork.Youneedtodividethemupnice.Wecanshow
youexamples.We'lltakethisguy.I'mgoingtodividethisupintotriangles.Youguystell
meifIdidthisright.
No.
Here'stheproblem,Iwantthetrianglestohelpmefindtheinteriorangles.These
trianglesandthisisn'tevenatriangle...Trianglesdonothavetheverticesatthe
quartersofthisshape,whichmeanstheverticesaregoingtobeabletoequalthe
cornersoftheshape.IwoulddoisIwanttodrawitnicely.I'mgoingtoshowyouguysa
tricksoyouguyscanusethisonyourdiagram.Thetrickis,pick1vertices.Highlight1
verticesontheshapethatyoujustdrew,doesn'tmatter...Anyone.Thenconnectthat
verticestoeveryothervertices.
Clearlytheadjacentonesarealreadyconnected,sonotnecessary.Forthese,connectit
andthenconnectit.Forthisone,Iguessthere'sonly1,connect.AlrightI'mgoingtofive
youguysaminute.Thiswillbemucheasierthantryingtoaddupallthestuff.Takeyour
2shapesandconnectonecornertoalltheothercornersthatyoucan.Thencountup
thenumberoftriangles.Giveitatry.Anytimeyoudrawit,itcancomeupkindofweird,
justmakesureitconnectstothecorners.[crosstalk00:22:24]Thethingis,thisoneis
alreadyconnected.[inaudible00:22:51]Idon'tknowtheanswersquiteyet.Youare
[inaudible00:23:03].
Allright,let'sfilloutourtable.Dependingonwhatyouguyshad,usethemall.Turnto
page150.Thisisthemuchbetterwayofinvestigatingthis,itdoesnotrequirethatyou
measureanyangles,justlikeatriangle.Itdoesnotrequirethatyoumeasureanyangles.
Let'sfillthisguyout.Numberofsides,verysimple.DoesitstopatoctagonordoIcut
thatoff?
Thelastone'sobtuse.
We'rejustgoingtoseethepatternyoucanextendthisasfarasyouguyswant.Now
numberoftrianglesformedto5diagonals.Foratriangle,Iassumethat's1.Howabout
foraquadrilateral?
2.
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2.Howaboutforapentagon?
3.
Hexagon?
4.
Nowwhatisthesumoftheinterioranglesof1triangle?
28.
Okay,soI'mjustgoingtodothat.Nowlet'sthinkaboutthis.Thetrianglesmakethe
polygon.Thetriangle'scornersmakethecornerofthepart.IfIaddupalloffthose
cornersandallthedifferentsides.Whichinfactisaddingupalloftheanglesinsidethe
triangles,Ishouldgetthefullsumoftheinteriorangles.Foratriangleobviouslyit's180.
Howaboutforaquadrilateral?
360.
Howaboutforapentagon?
540.
Okay,keepgoing.Filloutthetable.Followthepattern.Mucheasierthan
measurements.Bytheway,youguyswillbeusingthistablequiteabitfortoday,soyou
don'thavetocalculateiteverysingletime.Makesureit'sallfilledout.Don'twriteon
thisonebutyoucanuseit.[crosstalk00:25:29]Allrighthexagon?
720.
720.Hectagon?
900.
900.Octagon?
[crosstalk00:25:46]
Idon'tknowwhy.That'snotevenatime.WhatamIdoing?Let'stakethisaside.You
guysrecognizethesenumbers?Youeverseenthesebefore?Anyoneskateboard?
No.
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Youguysknowthesenumbers?TonyHawk,superfamousfordoinghe900,no?Ithink
onsnowboardsandstufftheydo1080'sandmore,somethinglikethat.Thisis
somethingthatissortoffamiliartomanyofyou,right?IntheXgames,orinthe
Olympicsorwhatever...Whensomeonespinsonaskateboardorsnowboardorski'sor
whatever,theydon'tusuallylandsideways.That'snotusuallyagoodwaytoland.They
areusuallylandingforwardorbackwards,whichisaspinof180.Ifyoukeepadding180,
aswe'vedonehere,yougetthosemultiples.Whatcomesafter1080?Whatcomesafter
1080?12?Theykeepgoing.Obviously,youcanmakethisgoonaslongasyouwant.
Youcanfindanyofthem.That'skindoftiteousjustadding180forever.IfIsaid,"All
right,Ihavethis100sidedpolygon.Howarewegoingtofindtheangles?"Kindofalot.
Let'sseeifwecanfigureoutabettermethodfordoingthis.
Here'swhatwejustdiduptothis.Youcanseethetriangles.Youcanseethesidesofthe
polygonsandthereissomemethodhere.Ifyouguyscouldanswerthosequestionson
thebottomofthatpage.Howmanytriangleswouldyouneedfora22sidedfigure?
Whatwouldtheanglesbe?Thenseeifyoucancomeupwithasimpleformulaforthis
andwe'lltalkaboutthisinasecond.
Whatquestions?
4-7Iguess.Allrightgiveitatry.
[inaudible00:27:39]
Ifthat'stheangleorifthat'sthetrick.Imeanseeifitworksforoneofthese.[crosstalk
00:27:48][crosstalk00:29:10]AllrightIwanttoseesomegoodequationshere.
[inaudible00:30:05]WhenIhad4sides,whatdidwedotothe4?[crosstalk00:30:22]
No,notnecessary.
Allright,letstakealook.Howmanyangles?
20.
20.Asyouguysnoticed,thepatternisminus2.There'sareasonforthisbutwedon't
havetogointoittoday.Thesumoftheinteriorangles?
3,600.
3,600right?Youtakethat20andyoumultiplyitbythe180.Let'srightdownour
theromhere,ofcourseyouwanttoputthisinyourbookafterwardssoyoucanuseit.
Let'sjustwriteitdownhere.Basically,thetheromforthesumoftheinterioranglesof
anypolygon...Theycallitan"N"sidedpolygon.IsawmanyofyouusingX,totally
acceptable.TheofficialoneisN.Itsays,thesummeasuresoftheinterioranglesofa
polygoninsidesisNminus2,keeptheparentheses,timesby180.
Thatgivesyouthenumberoftrianglesthenyoutimesitbythe180,whichistheangles
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insidethattriangle.Pleaseputitdownandthenlet'spractice.There'syourformulafor
theday.Nowyoucanseehowwegotit.Thatiswhatweusefortherestofourclass.I
shouldprobablyputthisontheboard.Allright,letsgivethisatry.Igotapolygonhere,
somepolygon.Let'sfindthemissingangle.[inaudible00:32:37]You'vegot30seconds.
Let'sseewhatyougot.Howmanysides?Let'sseewhatyougot?Allright,20seconds.
Allright.What'dweget?
64.
64.Whatkindofshapeisthis?
Quadrilateral.
Quadrilateral.Youcountthesides,thereforewecheckourtable.It's60degrees,addall
thoseupandthenminus360.Here'sanothertrickthatIliketodoguys.Sometimes,I
forgetthetable.SometimesIforgetthestuff,here'swhatIlearnedtodo.Iwilljust
literallydrawthetriangleonthepaper.NowIcansee,ohyeah,2triangles...180plus
180.Right?I'mdone.Thatisamuchnicerwaythanhavingtoreferenceatableor
memorizesomething.Justgenerateitforyourselfasyouformulateit.
OnelastthingtotalkaboutbeforeIletyouguyspractice.Wehavesomespecial
polygons.Wehavesomepolygonsthatarenice.Thesearecalledregularpolygons.A
regularpolygonisapolygonthatequalsides,equalangles.Weknowtheequilateral
triangle.Weknowthesquare.Ofcourse,alltheotherpolygonshavethesameproperty.
Theydon'thavefancynames.Wejustcallthemregularpolygons.Youcanseewehavea
bunchofdifferenttypes.They'reallequal.Obviouslythey'reanglesaregettingbiggeras
wego.Theyhavesomespecialpropertieshere.Themainpropertyistheiranglesare
equallikewesaid,whichmeanswecankindthemeasureofaspecificanglewithout
actuallyhavingtomeasurealltheotherangles.Thatworksoutprettygood.
Allright,let'sseeifwecanfigurethisout.Igot2polygonshere.Workwithyourtable
partners.Findmetheangleofthemeasureofasingleregularpolygon.Remember,they
areallcongruent...Verysimilartohowyoudoanequilateraltriangle.[crosstalk
00:36:07]Iseepeoplewhogottheansweralready.Useyourtables,reference.Allright,
Iseesomeanswersalready.Allright,Dustin?
72degrees.
72?
Igot72.
Howdidyouget72?
Idivided[inaudible00:36:44]
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Ohokay.You'reontherighttrackthough.You'reontherighttrack.Hehadasumand
divideditbyacertainnumberofsides.Ijustthinkhehasthewrongangles.Let'sfixit.
Whatisthesumofthistypeofshape?
540.
540andwedivideitby?
5.
5becauseit's5sidesandthey'reallequal.Thatgivesus?
108.
108,verygood.108shouldbeforA.ForB,somethingyouhaddown?
135.
135.Whateverthisiswitha1080,somethinglikethat.Thataddsupcloseto1080and
youdivideitby8.Thisistheotherthingyouguyswillsee.Theywillnotgiveyouany
angles.Theywillgiveyouashapeandyouneedtodothedivisionyourselftofindthose
angles.Thesearekindofnice.Inthebooktheyhavethistable.I'mtryingtothinkifwe
havetimetodothis.YouknowIthinkwe'regoingtoskipthisfornow.Youcangenerate
themasyougo.Wemayhavetimeattheendtofillthisout.Idon'twanttodoitright
now.
Basically,theywantyoutokindofgeneratethoseinteriorangles.Let'sjustputafewof
theminheresinceyouguysknowsomeofthem.Wesaidthe8onewas...Wasit135?
Forthe8sidedone?Wejustsaid135?Thenthepentagonwas108.Howabouta4sided
shape?Youguysknowthisone.Whatdowecallaregularquadrilateral?What'sthe
fancyname?
[inaudible00:38:14]
Whatdowecalltheshape?
90degree.
Theyare90.Whatdowecallthatshape?
Isn'titasquare?
Ofcourse,asquare.Thenwehavethetriangle.Youguysknowthatone.
180.160.
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Again,don'tworryabouttherestoftheseyoucangeneratethemasneeded.We'lluse
themlater.Ifyouwanttofillthisoutwhenwearedoingourpacketthat'sfine.Ijust
wanttomoveonforasecondnow.Aswegoforwardyoucanfinditeachtime.Basically
youjusthavetofindthatsumandthendivideitbyN,whateverthenumberofsidesare
forthenumber.Iwanttotalkaboutonelastthing.Isawsomeofyouguysdrawing
thingsthatlookedlikethis.Isthisapolygon?
No.
That'sfine.It'sapolygon.Howisitdifferentthanthenormalkindthatwedraw?
Somethingabouttheshape,yeah?Somethingaboutit'sanglesactually.Ithasthese
reflexanglesright?Thereflexanglesarebiggerthan180andthey'reinteriorlikethat.
They'rekindofhardtomeasure.They'restillangles.Thisisstillapolygon.Thisisnota
polygon.Whenyoudrawitsoitoverlapsthecorners.Thatsuddenlybecomesseparate
polygons.Apolygonmusthaveacontinuousarea,whichmeansthatalltheareahere
mustbeconnectedtoitself.
I'mgoingtoshowyouguysalittleanimationjusttoprovetoyouguysthatthisworked.
WhatI'vealwayslearnedorwhatthey'vealwayssaid...Iwaslookingatatextbook,they
alwayssayusethesepolygonsright?Usethepolygon.Don'tusetheconcaveones.I'm
like,"Wellwhynot?It'sstillapolygon."Ifoundsomeprogramthat'sgoingtoanimate
this.Let'ssee.Let'sseeifitworks.Iwantyouguystolookdownhere.Thesearethe
anglesthatthecomputerisjustmeasuringthemandthenit'saddingupthose3
numbers,solet'sseewhat'shappeninghere.
Thatangleischanging.What'shappeningtothesum?Rocksolid.Youcanmakethem
reallytiny,reallybig.Weknewthat,trianglesobviously.Let'slookatanothertypeof
shape.Howaboutthisguy?Youcanseethesum,540asweknow.Let'smessitup,
changeit.Evenifwemakeitcrazylikethis,rocksolid.Thisisthecoolthing,itdoesn't
evenmatterwhattheshapelookslike,unless...Thatworkstoo.Unless,youoverlapit
likethisandnowsuddenlyitcannotcalculatebecauseit'stootonic.Youguyscansee.
Ofcourse,foranyshapethiswillwork.Here'sthe1080,the8sidedshape...MakePAC
Man...Anycrazyshapethatyouwant.Alltheanglesmovebutyoucanseeasoneof
themgetsbigger,itsortofforcestheotheronetogetsmaller,nomatterwhatcrazy
shapeyoumake.Itwillstillwork,unlessofcourseyoucrossoneofthemtonotmakea
polygon.Idon'trememberifIdidthat.Youguyscanseeeventhatonewillwork.
Usuallytheydon'tgiveyouonesthatlooklikethisbutyouguygettheidea.Itstillwould
work.Okay.We'regoingtodo2morethingshere.Guesswhoneedshelp?Makeita
littlebit.Yourfavorite.Allrightso[inaudible00:41:41],gottogowiththegazebo.You
guysremembertheshapeofthegazebo?Isitahexagon?It'sahexagon.Ifshewantsa
benchtogointhemiddletokindofmatchthatgazebo,here'sthebench.Youhaveto
cutthepiecesofwoodtokindofnailtogetherhere.WithyourtablepartnersIwantyou
tofindmethisangle.Whatanglewouldhavetocutthatsothese2thingswouldjoin
nicely?Youguysshouldgetthisprettyquickactuallysinceyouknowalltheotherstuff.
Whatisthissortofblack...Whatisthisangleherethatcutdownthesize?Drawthe
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hexagonfromthetopifyouneedto.Seewhatyouguysgot.
Hexagonshapes.I'mgoingtopassoutsomethingwhileyouguysaredoingthisjustkeep
itonyourdesk.Maybefindthis.Thismaybeastrategy.Whatyoufoundisthisangle.All
rightStudent?
60.
Howdidyouget60?
Divided720.
Okay,that'safullinteriorangle.
Imean6by7feet.
Noyouwereright.Youtookthe720whichistheinterioranglesofthehexagon.You
divideditby6.Thatgivesyouthisangle.Thisoutsideangle.That'snottheanglewe're
cuttingwithoursaw.How'dyougetheoctagon?
Idivideditby2.
Rememberwetalkedaboutthis.Whenyou'redoingtrimorwhateverinahouse,you
watchedthatvideo?Youwanttomakeitequalnomatterwhattheangleisyouhaveto
dissectit.Theeasiestwayofcourseistoknowtheangleanddivideit.Rememberwe
hadthosepapertrigs,thosecompasstrigsandeverything?Thatoneworkstoo.
Thisisareallifeapplication.I'vehadtodostufflikethiswhenIambuildinglittlehome
projectsandstuffandIofcourseappreciateyourguyshelp.Onelastpiece,whatifwe
have2polygonsthatarestucktogether?Let'sdothisonetogetherandthenwe're
goingtodosomegroupworkandpresentafewproblemsattheend.Let'sfigureout
whathappenedhere.Wegot2regularpolygons.Wehaveasquareandahexagon.We
wanttofindthisangle.Thissortofoutsidegroupangle.WhatshouldIlookforfirst?
Theinteriorangle.
Theinteriorangleofcourse.Theotherside,rightHowmuchforthesquare?
90.
Allright,sowegot90degreesrighthere.Howaboutforthehexagon?Wejustdidthis
one.
120.
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Allright,sowhat'snext?
210.
How'dyougetthat?
Youaddthemup.
Okay,soyouaddedup90plus120.Okayandyougot210.That'sthisangleandand
thenwhat?
150.
How'dyouget150?
[crosstalk00:45:50]360.
Youhad360minus210andyougot?
150.
Thisistheotherskillandwetalkedaboutthislastclass.Manytimesyoucanfindpartof
acircleandthenyoucanusealittlesubtractiontofindtheothermissingpartofthe
circle.Justlikewhenwehadalinearpair,youcanfindthatmissinganglethataddsup
to180.Theseanglesaddupto360.Youcansortofgeneratethatbysubtractingitoff
theend.Thesearethebasicskillsthatweneedtobesuccessfulatthis.We'regoingto
besolvingsomeproblemssograbthepaperinfrontofyouandIknowit'skindofsmall
soifyouneedroom,pleaseuseyourbook.Thisisgoingtobeourpractice.Let'stake...
What'dIsay.
We'lldoabout15or20minutes.We'llseehowlongittakes.Idon'tknow.There'sa
coupleproblemsonhere,therearesomeequations,sobecarefultowritethefull
equation.Iknowyouguysaregoodatsortoffindingsomeofthestuffinyourhead,but
ifyoucouldwritethefullequationoutinyourbookthatmayhelpyoufortheoneswith
alotofextrasinthem.Justbecarefulwiththat.Alright,I'llturnontheradio.Wehave
15minutesletssay.Pleaseworkonyourhandouts.Pleaseworkquietlywithyourtable
partnersandbereadytopresentacoupleattheendtoputthemupontheboard.
[crosstalk00:48:04][crosstalk00:50:02][crosstalk00:52:05][crosstalk00:54:06]
[crosstalk00:56:02]Let'ssay14moreminutes.Helpeachotherout.We'llpresenta
coupleoftheseatthatpoint.[crosstalk00:58:04][crosstalk01:00:16]Justremember
guys,youcanalwayscheckyouranswerattheend.Onceyougetananglemakesureif
it'sobtuse,isitacute?Doesitmakesensewiththetypeofpolygonyouweregiven?
[crosstalk01:02:03][crosstalk01:04:02][crosstalk01:06:03][crosstalk01:08:02]We'll
takeabout4moreminutes.Ifyou'refinishedyoucanstartyourhomework.We'll
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presentafewoftheseandtalkaboutthemintheend.[crosstalk01:10:02][crosstalk
01:12:03]Youguyshave10minutes.
OnequickthingIwasn'ttoadd.Idon'tknowifthisishowit'ssupposetobe...The
answer.YouguyscanzoominonC.C...Idon'tknowifthere'senoughinformationto
solvethisone.ItshouldbebutIfeellikeI'mmissingsomething.Justtomakethiswork,
theywaythatwe'vetalkedaboutit...Ifyoucouldmakethese2lineparallel,thenyou
cansolveit.Right?Thenyouhavemoreequivalenciesthatyoucanthenwrite,yeah?If
youhavethesethenthisangleiscongruenttothisangle.Thiswouldbe180minus3y.
Suddenly,youhaveenoughvariablestosolveforthis.That'swhatIwouldsayjustto
makealittlebitmoredoable.
I'mnotexactlysurewhattheywantedustodoforthisone.Theotheronescameout
prettynice.Ifeellikewe'remissingsomethingforthatonethough.Imadeamistake
whenImadethecopiesIdon'tknow.Let'sgiveit1moreminute.Finishupyourlast
problem.We'regoingtopresentafewoftheseontheboard.We'regoingtotalkabout
themandthenwewillour...Ifyoucouldyourprotractorsbackonthestickynotesfor
thenextperiodthatwouldbeawesome.Ithinkthere's2oneachone.Ifyoucoulddo
that,thatwouldbeawesome.[crosstalk01:14:35]Alittlemoretime.
Allrightguys.Let'stalkaboutit=atthefront.I'llcheck...ZoomintoBforasecond.Did
wedothatone?Wasthatoneofmyexamplesonthenotes?Helpmefindit.Allright,a
6sidedshape.What'safancynameforthat?
Hexagon.
Thereare6sides,youfound720,whichwasthesum.Howdidyouknowyoucould
divideby6?Why6?
Ithas6sides.
Youcannotdothatforexamplebecause-
[inaudible01:16:38]Askmeagain.
Thetheromhastodowitharegularhexagon.Howdoweknowthisisaregular
hexagon?
Theanglesarealleven.
Yes,theanglesarealleven.Thisactuallydoesn'thavetobearegularhexagon.Icouldif
Ireallywantedtostretchthismiddleside-[crosstalk01:17:02]Icouldstretchthis
middlesideouttobetwiceaslongandtheanglesdonotchange.Thisisnotnecessarily
aregularhexagon.Theregularnessthatcounts,theanglesumsright?Ortheangle
equivalencies,thosearethere,sowedon'treallyneedtoworryaboutthesize.It
doesn'thaveasizelabeledsoit'snotnecessarilyaregularhexagon.[inaudible01:17:24]
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Zoomin.
Allrightlet'stakealookattheguide.Thisismoreofthehomeworkquestionsthatyou
guyswillsee.Basically,thereareabunchofvariablesinsidethispolygon.Let'sseeifwe
canfigureoutwhatishappening.Firstofall,whattypeofpolygonisthis?
[inaudible01:17:50]6.
6sides,soHexagon.Isthisaregularhexagon?
No.
Iassumenot,nothingislabeled.Whatdidyoudo?
Iaddedallthesidesandaddeduptheequationandmadethe[inaudible01:18:04]720.
That'sthesum.
That'sthesumoftheinteriorhexagon.
Addedtogetheris52Xtimes8.Youadd8tobothsides,28.Thenyoudivideby52and
get14.
Excellent.Xis14,butthatisnotanyoftheangles.Howdidyoufindtheactualangles
afterthat?
Iplugged14intoX.
Verygood.Didyoucheck?Doalltheseaddupto720?
No.
That'ssomethingyoucoulddoagain.Itrustyourwork.Thatissomethingyoucoulddo
tofindthetotalsumtoseeifyouransweractuallymakessense.NowacouplethingsI'll
pointoutherethatIreallylike.Ireallylikethatyouhavethefullequation,notjust
piecesofit.Youcankindofkeepyourselforganized.Ireallylikethatyoukept
everythinginparentheses,soyoudidn'thavetoworryaboutnegativesignsorstuff
spillingout.Youguyscanseecombiningtheliketerms,right?YouwanttogetalltheX's
togethersoyoucandivideoffthatconditionandofcoursewemovethe8first,divideit
anditcomesoutnice.Iguessitdoesn'thavetocomeoutnice,it'samathproblemina
textbook,soIassumeit'sgoingtobeokay.
Allright,awesomeguys.Dowehaveonemoreorarewegood?Actually,wearealmost
outoftime.Let'sjustgothroughthebottomhere.Thesearetheanswers.Let'ssee
whatwegot.Let'szoom...Oh,therewego.Allright,youguyscanseemostofthese
anglessuchasD,A,C,E,theseareinteriorangleswe'vealreadycalculated.Thoseare
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trig.Someoftheotheronesthough,youhavetodoalittlebitmoremath.Forexample,
I...We'vegotthis60degreeanglerighthereinthisequilateraltriangleandIissortof
thatoutsideanglewhichissortofinbetweentheoctagonangleandthetriangleangle.
Youkindofjustdoalittlesubtraction.ThisisatrickIwantyouguystobethinking
about.Wementionedthislasttime.
Manytimeswhenyouarefindingamissingangle,theywillnotlabelthethingthatyou
actuallyneedtofind.TheywantyoutofindI,buttheydon'ttellyou,youhavetofindX
overherefirst.YougottofindthebigangleYsecond.Theydon'ttellyouthat.Youhave
tosortoffindwhatyouwant.Thisisabalancingact,becauseyoucouldfind50different
anglesand49ofthemwouldnotbeusefultoyou.Youcouldfindthisangleandthis
angle,butthey'renotreallyuseful.Youwanttoconcentratearoundtheareathatyou
actuallyhave.Awesomeguys,grabthishandout.Youcanholdthisfornow,finishitupif
youhavenot.
Grabyournotes,let'stakeourtest.Actually,ifyoucouldcloseyourworkbook,Iwant
toseeifyouguyscandothiswithoutlookingatthetable.Justmakesureyouputyour
calculatorsbackattheend.Ifanyoneneedstolookattheirtest,comeseemeafter
class.Areyouready?Perfecttiming.Whatisthisshapecalled?
Onogom.
Onogon?[crosstalk01:22:09]Howmanysides?
9.
9.Howmanytriangles?
7.
7.Howmanyinterioranglestotal?Right.Thesumoftheinteriorangles?
60.
Thesumorthedivision?
140.
140verygood.Thisisinterioranglesofpolygons.Weknowhowtofindthiswithouta
table.Wecandivideupourshapeintotrianglesandthenaddupthesumofthose
triangles.The[inaudible01:23:06]ofsomethingyouguysalreadyknow.Allrightthanks
verymuchguys.Putyourprotractorsonyourcards,calculators.Haveawonderful
weekend.Comeseemeifyouwanttoseeyourtest.Ifyouhaven't,youneedtotake
yourtest.[crosstalk01:23:26]
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