Name:____________________________________________________Date:________Period:_______
UsingFormalLanguage
PartA:
Yesterday,weconstructed6trianglesgiven3sidelengths.Sallywasabsentyesterday,again.Sallysays,
“Iconstructedthetwotrianglesbelow.Theyarenotcongruentbecausetheydon’tevenlookalike.”
1.ExplaintoSallyhowyouknowthetrianglesarecongruent.
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Therearefourwaysthatwehavelearnedsofartoshowthattwotrianglesarecongruent.Thefour
waysaregivenbelow.Howcanyouuseeachofthefollowingtoprovetwotrianglesarecongruent?
a) ThroughRigidMotions___________________________________________________________
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b) 𝑆𝑆𝑆 ≅ 𝑇ℎ𝑒𝑜𝑟𝑒𝑚________________________________________________________________
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c) 𝑆𝐴𝑆 ≅ 𝑇ℎ𝑒𝑜𝑟𝑒𝑚________________________________________________________________
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d) 𝐴𝑆𝐴 ≅ 𝑇ℎ𝑒𝑜𝑟𝑒𝑚________________________________________________________________
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IntroductiontoProofs:FormalLanguage
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Name:____________________________________________________Date:________Period:_______
PartB:
DefinitionofProof–evidencesufficienttoestablishathingastrue
MathematicalProof-anargumentthatbeginswithknownfacts,proceedsfrom
therethroughaseriesoflogicaldeductions,andendswiththestatementyou’re
tryingtoprove.
2.WhatwasitthatyouweretryingtoprovetoSally?______________________
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3.Whatfactsdidyouknowaboutthetrianglesthatcouldbeusedasevidence?
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Aparagraphproofisonewayaproofisoftenwritten.Theadvantageofa
paragraphproofisthatyouhavethechancetoexplainyourreasoninginyour
ownwords.Inaparagraphproof,thestatementsandtheirjustificationsare
writtentogetherinalogicalorderinaparagraphform.Thereisalwaysadiagram
andastatementofthegivenandprovesectionsbeforetheproof.
Given:
Prove:
FromtheGiven,Iknow_______________________________________________.
Therefore,triangle_____and_____mustbecongruentbecause_____________
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IntroductiontoProofs:FormalLanguage
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Name:____________________________________________________Date:________Period:_______
PartC:
4.Provethatthetwotrianglesarecongruent.UsingaTriangleCongruenceTheorem.
Given:
Prove:
FromtheGiven,Iknow_______________________________________________.
Therefore,__________________________________________________________
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5.Provethatthetwotrianglesarecongruent.UsingRigidMotions.
Given:
Prove:
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IntroductiontoProofs:FormalLanguage
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Name:____________________________________________________Date:________Period:_______
TeacherDirections
ContentObjective:Studentswilluseparagraphproofstoprovetwotrianglesare
congruent.
Materials:
Handout–1perstudent
Directions:
PartA.
StudentscansaythatSallyneedstogetapieceofpattypaperanduseasequence
ofrigidmotionstomaponfigureontotheotherfigure.Studentsmayalsostate
thetrianglesarecongruentbecauseofSSScongruencetheorem.Eitheransweris
sufficientrightnow.
Fourwaystrianglesarecongruent:
a) RigidMotions–wemustshowthatthereisasequenceoftranslations,
reflectionsand/orrotationsthatmapsonefigureontotheother
b) SSS–wemustshowthat3pairsofcorrespondingsidesarecongruent.
c) SAS-wemustshowthat2pairsofcorrespondingsidesandthe
correspondingincludedanglesarecongruent
d) ASA–wemustshowthat2pairsofcorrespondinganglesandthe
correspondingincludedsidesarecongruent
***Makesurethatstudentssayincludedsideoranglewhenneeded.Youmay
havetodemonstratethedifferencebetweenincludedangle/non-includedangle.
IntroductiontoProofs:FormalLanguage
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Name:____________________________________________________Date:________Period:_______
PartB.
Question2&3aretoshowstudentsthattheyarealreadyfamiliarwith
identifyingthegivenandwhattheyweretheytryingtoproveinSally’sproblem.
Havestudentsfillintheblanks.
Given:𝐺𝐼 ≅ 𝐹𝐷, 𝐼𝐻 ≅ 𝐷𝐸, 𝐻𝐺 ≅ 𝐸𝐹 Prove:∆𝐺𝐼𝐻 ≅ ∆𝐹𝐷𝐸
Since𝐺𝐼 ≅ 𝐹𝐷, 𝐼𝐻 ≅ 𝐷𝐸, 𝐻𝐺 ≅ 𝐸𝐹wasgiven,thereforeIknowthatthetwo
triangles∆𝐺𝐼𝐻 ≅ ∆𝐹𝐷𝐸arecongruentbecauseof𝑆𝑆𝑆 ≅ 𝑇ℎ𝑒𝑜𝑟𝑒𝑚
PartC.
UsingaTriangleCongruenceTheorem
• FollowsexactlylikePartBbutwithASA≅theorem.
UsingRigidMotions.
- Answersmayvarybutneedstoincludeat-leastonetranslationandone
reflection.Makesurestudentsexplainwheretheyaretranslatingtoand
whatlinetheyarereflectingover.
IntroductiontoProofs:FormalLanguage
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