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Transcript
24
7. 5 Congruent Triangles
to the Rescue
A Practice Understanding Task
Part1
ZacandSioneareexploringisoscelestriangles—trianglesinwhichtwosidesarecongruent:
Zac:Ithinkeveryisoscelestrianglehasalineofsymmetrythatpassesthroughthevertex
pointoftheanglemadeupbythetwocongruentsides,andthemidpointofthethirdside.
Sione:That’saprettybigclaim—tosayyouknowsomethingabouteveryisoscelestriangle.
Maybeyoujusthaven’tthoughtabouttheonesforwhichitisn’ttrue.
Zac:ButI’vefoldedlotsofisoscelestrianglesinhalf,anditalwaysseemstowork.
Sione:Lotsofisoscelestrianglesarenotallisoscelestriangles,soI’mstillnotsure.
1. WhatdoyouthinkaboutZac’sclaim?Doyouthinkeveryisoscelestrianglehasalineof
symmetry?Ifso,whatconvincesyouthisistrue?Ifnot,whatconcernsdoyouhaveabout
hisstatement?
2. WhatelsewouldZacneedtoknowaboutthecreaselinethroughinordertoknowthatitisa
lineofsymmetry?(Hint:Thinkaboutthedefinitionofalineofreflection.)
3. SionethinksZac’s“creaseline”(thelineformedbyfoldingtheisoscelestriangleinhalf)
createstwocongruenttrianglesinsidetheisoscelestriangle.Whichcriteria—ASA,SASor
SSS—couldheusetosupportthisclaim?Describethesidesand/oranglesyouthinkare
congruent,andexplainhowyouknowtheyarecongruent.
4. Ifthetwotrianglescreatedbyfoldinganisoscelestriangleinhalfarecongruent,whatdoes
thatimplyaboutthe“baseangles”ofanisoscelestriangle(thetwoanglesthatarenot
formedbythetwocongruentsides)?
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SECONDARY MATH I // MODULE 7
25
SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
5. Ifthetwotrianglescreatedbyfoldinganisoscelestriangleinhalfarecongruent,whatdoes
thatimplyaboutthe“creaseline”?(Youmightbeabletomakeacoupleofclaimsaboutthis
line—oneclaimcomesfromfocusingonthelinewhereitmeetsthethird,non-congruent
sideofthetriangle;asecondclaimcomesfromfocusingonwherethelineintersectsthe
vertexangleformedbythetwocongruentsides.)
Part2
LikeZac,youhavedonesomeexperimentingwithlinesofsymmetry,aswellasrotational
symmetry.InthetasksSymmetriesofQuadrilateralsandQuadrilaterals—BeyondDefinitionyou
madesomeobservationsaboutsides,angles,anddiagonalsofvarioustypesofquadrilateralsbased
onyourexperimentsandknowledgeabouttransformations.Manyoftheseobservationscanbe
furtherjustifiedbasedonlookingforcongruenttrianglesandtheircorrespondingparts,justasZac
andSionedidintheirworkwithisoscelestriangles.
Pickoneofthefollowingquadrilateralstoexplore:
•
Arectangleisaquadrilateralthatcontainsfourrightangles.
•
Arhombusisaquadrilateralinwhichallsidesarecongruent.
•
Asquareisbotharectangleandarhombus,thatis,itcontainsfourrightanglesand
allsidesarecongruent
1. Drawanexampleofyourselectedquadrilateral,withitsdiagonals.Labeltheverticesofthe
quadrilateralA,B,C,andD,andlabelthepointofintersectionofthetwodiagonalsaspointN.
2. Basedon(1)yourdrawing,(2)thegivendefinitionofyourquadrilateral,and(3)information
aboutsidesandanglesthatyoucangatherbasedonlinesofreflectionandrotational
symmetry,listasmanypairsofcongruenttrianglesasyoucanfind.
3. Foreachpairofcongruenttrianglesyoulist,statethecriteriayouused—ASA,SASorSSS—to
determinethatthetwotrianglesarecongruent,andexplainhowyouknowthattheangles
and/orsidesrequiredbythecriteriaarecongruent(seethefollowingchart).
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SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
Congruent
Triangles
CriteriaUsed
(ASA,SAS,SSS)
IfIsayΔRST≅ΔXYZ
basedonSSS
HowIknowthesidesand/orangles
requiredbythecriteriaarecongruent
thenIneedtoexplain:
• howIknowthat RS ≅ XY ,and
• howIknowthat ST ≅ YZ ,and
• howIknowthat TR ≅ ZX soIcanuseSSScriteriatosayΔRST≅ΔXYZ
4. Nowthatyouhaveidentifiedsomecongruenttrianglesinyourdiagram,canyouusethe
congruenttrianglestojustifysomethingelseaboutthequadrilateral,suchas:
• thediagonalsbisecteachother
•
thediagonalsarecongruent
•
thediagonalsareperpendiculartoeachother
•
thediagonalsbisecttheanglesofthequadrilateral
Pickoneofthebulletedstatementsyouthinkistrueaboutyourquadrilateralandtryto
writeanargumentthatwouldconvinceZacandSionethatthestatementistrue.
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CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
7. 5 Congruent Triangles to the Rescue – Teacher Notes
A Practice Understanding Task
Purpose:Thepurposeofthistaskistoprovidestudentswithpracticeinidentifyingthecriteria
theymightuse—ASA,SASorSSS—todetermineiftwotrianglesembeddedinanothergeometric
figurearecongruent,andthentousethosecongruenttrianglestomakeotherobservationsabout
thegeometricfiguresbasedontheconceptthatcorrespondingpartsofcongruenttrianglesare
congruent.Asecondarypurposeofthistaskistoallowstudentstocontinuetoexaminewhatit
meanstomakeanargumentbasedonthedefinitionsoftransformations,aswellasbasedon
propertiesofcongruenttriangles.Thefocusshouldbeonusingcongruenttrianglesand
transformationstoidentifyotherthingsthatcanbesaidaboutageometricfigure,ratherthanonthe
specificpropertiesoftrianglesorquadrilateralsthatarebeingobserved.Theseobservationswillbe
moreformallyprovedinSecondaryII.Theobservationsinthistaskalsoprovidesupportforthe
geometricconstructionsthatareexploredinthenexttask.
CoreStandardsFocus:
G.CO.7Usethedefinitionofcongruenceintermsofrigidmotionstoshowthattwotrianglesare
congruentifandonlyifcorrespondingpairsofsidesandcorrespondingpairsofanglesare
congruent.
G.CO.8Explainhowthecriteriafortrianglecongruence(ASA,SAS,andSSS)followfromthe
definitionofcongruenceintermsofrigidmotions.
SeealsoMathematicsInoteforG.CO.6,G.CO.7,G.CO.8:Rigidmotionsareatthefoundationofthe
definitionofcongruence.Studentsreasonfromthebasicpropertiesofrigidmotions(thatthey
preservedistanceandangle),whichareassumedwithoutproof.Rigidmotionsandtheirassumed
propertiescanbeusedtoestablishtheusualtrianglecongruencecriteria,whichcanthenbeusedto
proveothertheorems.
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RelatedStandards:G.CO.10
StandardsforMathematicalPracticeofFocusintheTask:
SMP3–Constructviableargumentsandcritiquethereasoningofothers
SMP7–Lookforandmakeuseofstructure
AdditionalResourcesforTeachers:
Acopyoftheimagesusedinthistaskcanbefoundattheendofthissetofteachernotes.These
imagescanbeprintedforusewithstudentswhomaybeaccessingthetaskonacomputerortablet.
TheTeachingCycle:
Launch(WholeClass):
Makesurethatstudentsknowthedefinitionofanisoscelestriangleandgivethemseveralisosceles
trianglestofold—essentiallyrecreatingZac’spaper-foldingexperimentasdescribedinpart1ofthe
task(seeattachedhandoutofisoscelestriangles).Askstudentsiftheyseeanycongruenttriangles
insideofthefoldedisoscelestriangle,andwhatcriteriaforcongruenttriangles—ASA,SASorSSS—
theycouldusetoconvincethemselvesthattheseinteriortrianglesarecongruent.Workthroughthe
additionalquestionsinpart1withtheclass,givingstudentstimetothinkabouteachquestion
individuallyorwithapartner.
HelpstudentsseethedifferencebetweenverifyingZac’sclaim(“everyisoscelestrianglehasalineof
symmetrythatpassesthroughthevertexpointoftheanglemadeupofthetwocongruentsides,and
themidpointofthethirdside”)throughexperimentation—paperfolding—andajustificationbased
ontransformationsandcongruenttrianglecriteria.Itappearsfromfoldingonesideoftheisosceles
triangleontotheotherthattwocongruenttrianglesareformed.ThiscanbejustifiedusingtheSSS
trianglecongruencecriterion:thelinethroughthevertexandthemidpointoftheoppositesideis
commontobothinteriortriangles(S1);themidpointoftheoppositesideformstwocorresponding
congruentsegmentsintheinteriortriangles(S2);andbydefinitionofanisoscelestriangletheother
pairofsidesintheinteriortrianglesarecongruent(S3).Sincetheinteriortrianglesarecongruent
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SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
bySSS,wecanalsoconcludethatthethreecorrespondinganglesarecongruent.Thisleadstosuch
additionalpropertiesas:thebaseanglesoftheisoscelestrianglearecongruent;thevertexangleis
bisectedbythelinethroughthevertexandmidpointoftheoppositeside;andthelinethroughthe
vertexandmidpointoftheoppositesideisperpendiculartothebasesincetheanglesformedare
congruentandtogetherformastraightangle.Collectively,thesestatementsjustifyZac’sclaimthat
everyisoscelestrianglehasalineofsymmetry.
Explore(SmallGroup):
Theguideddiscussionofpart1ofthistaskwillpreparestudentstoworkmoreindependentlyon
part2.Youmaywanttoassigndifferentgroupstoaparticularquadrilateral,soallofthe
quadrilateralsgetexplored.Centertheexplorationtimeonpart2,questions2and3—lookingfor
congruenttriangles,andlistingthecriteriathatwasusedtoclaimthatthetrianglesarecongruent.
Fastfinisherscanworkonpart2,question4—justifyingotherpropertiesofquadrilateralsbasedon
correspondingpartsofcongruenttriangles.
Discuss(WholeClass):
Thefocusofthediscussionshouldbeonpart2,question2—identifyingcongruenttrianglesformed
indifferenttypesofquadrilateralsbydrawinginthediagonals.Asstudentsclaimtwotrianglesare
congruent,askthemtoexplainthetrianglecongruencecriteria—ASA,SASorSSS—theyusedto
justifytheirclaim.Astimeallows,discusssomeoftheotherclaimsthatcanbemadeaboutthe
quadrilateralsbasedoncorrespondingpartsofcongruenttriangles.
AlignedReady,Set,Go:Congruence,ConstructionandProof7.5
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SECONDARY MATH I // MODULE 7
7.5
CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
READY, SET, GO!
Name
PeriodDate
READY
Topic:Transformationsoflines,connectinggeometryandalgebra.
Foreachsetoflinesusethepointsonthelinetodeterminewhichlineistheimageandwhichis
thepre-image,writeimagebytheimagelineandpreimagebytheoriginalline.Thendefinethe
transformationthatwasusedtocreatetheimage.Finallyfindtheequationforeachline.
1.
2.
N'
M'
N
M
a.DescriptionofTransformation: b.Equationforpre-image:
c.Equationforimage:
Useforproblems3thorugh5.
a.DescriptionofTransformation:
b.Equationforpre-image:
c.Equationforimage:
3.a.DescriptionofTransformation:
b.Equationforpre-image:
c.Equationforimage:
4.Writeanequationforalinewiththesameslope
thatgoesthroughtheorigin.
5.Writetheequationofalineperpendicularto
theseandthoughthepointO’.
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SECONDARY MATH I // MODULE 7
7.5
CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
Afterworkingwiththeseequationsandseeingthetransformationsonthecoordinategraphitisgood
timingtoconsidersimilarworkwithtables.
6.Matchthetableofvaluesbelowwiththeproperfunctionrule.
I
II
x
-1
0
1
2
f(x)
16
14
12
10
III
x
-1
0
1
2
A.! ! = −! ! − ! + !
B.! ! = −! ! − ! + !"
C.! ! = −! ! − ! + !
f(x)
14
12
10
8
IV
x
-1
0
1
2
f(x)
12
10
8
6
V
x
-1
0
1
2
D.! ! = −! ! + ! + !
E.! ! = −! ! + ! + !"
f(x)
10
8
6
4
x
-1
0
1
2
SET
Topic:UseTriangleCongruenceCriteriatojustifyconjectures.
Ineachproblembelowtherearesometruestatementslisted.Fromthesestatementsa
conjecture(aguess)aboutwhatmightbetruehasbeenmade.Usingthegivenstatementsand
conjecturestatementcreateanargumentthatjustifiestheconjecture.
7.Truestatements: Conjecture: ∠A ≅ ∠C PointMisthemidpointof!"
∠!"# ≅ ∠!"#
a.Istheconjecturecorrect?
!" ≅ !"
b.Argumenttoproveyouareright:
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f(x)
8
6
4
2
29
SECONDARY MATH I // MODULE 7
7.5
CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
8.Truestatements
∠ !"# ≅ ∠ !"#
!" ≅ !"
9.Truestatements
∆ !"#isa180°
rotationof∆ !"#
Conjecture:!"bisects∠ !"#
a.Istheconjecturecorrect?
b.Argumenttoproveyouareright:
Conjecture:∆ !"# ≅ ∆!"#
a.Istheconjecturecorrect?
b.Argumenttoproveyouareright:
GO
Topic:Constructionswithcompassandstraightedge.
10.Whydoweuseageometriccompasswhendoingconstructionsingeometry?
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SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
Performtheindicatedconstructionsusingacompassandstraightedge.
11.Constructarhombus,usesegmentABasonesideandangleAasoneoftheangles.
12.ConstructalineparalleltolinePR
andthroughthepointN.
13.ConstructanequilateraltrianglewithsegmentRSasoneside.
14.Constructaregularhexagoninscribed
inthecircleprovided.
15.ConstructaparallelogramusingCDasoneside
andCEastheotherside.
16.BisectthelinesegmentLM.
17.BisecttheangelRST. Mathematics Vision Project
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7.5