Types of angles vertical,complementary and supplementry angles
Lesson 9 Claims and Conjectures · Vertical Angles When two lines intersect, the opposite angles...
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SECONDARY MATH II // MODULE 5 GEOMETRIC FIGURES – 5.5 Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org Lesson 9 Claims and Conjectures A Solidify Understanding Task The diagram from How Do You Know That? has been extended by repeatedly rotating the image triangles around the midpoints of their sides to form a tessellation of the plane, as shown below. Using this diagram, we will make some conjectures about lines, angles and triangles and then write proofs to convince ourselves that our conjectures are always true. CC BY Denise Krebs https://flic.kr/p/gG6fYo Page 69
Transcript of Lesson 9 Claims and Conjectures · Vertical Angles When two lines intersect, the opposite angles...
SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.5
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Lesson 9 Claims and Conjectures
A Solidify Understanding Task
ThediagramfromHowDoYouKnowThat?hasbeenextendedbyrepeatedlyrotatingtheimage
trianglesaroundthemidpointsoftheirsidestoformatessellationoftheplane,asshownbelow.
Usingthisdiagram,wewillmakesomeconjecturesaboutlines,anglesandtrianglesandthenwrite
proofstoconvinceourselvesthatourconjecturesarealwaystrue.
CC
BY
Den
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SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.5
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
VerticalAngles
Whentwolinesintersect,theoppositeanglesformedatthepointofintersectionarecalledvertical
angles.Inthediagrambelow,∠1and∠3formapairofverticalangles,and∠2and∠4form
anotherpairofverticalangles.
Examinethetessellationdiagramabove,lookingforplaces
whereverticalanglesoccur.(Youmayhavetoignoresome
linesegmentsandanglesinordertofocusonpairsofvertical
angles.Thisisaskillwehavetodevelopwhentryingtoseespecificimagesingeometricdiagrams.)
Basedonseveralexamplesofverticalanglesinthediagram,writeaconjectureaboutverticalangles.
Myconjecture:
ExteriorAnglesofaTriangle
Whenasideofatriangleisextended,asinthediagrambelow,theangleformedontheexteriorofthe
triangleiscalledanexteriorangle.Thetwoanglesofthetrianglethatarenotadjacenttotheexterior
anglearereferredtoastheremoteinteriorangles.Inthediagram,∠4isanexteriorangle,and∠1
and∠2arethetworemoteinterioranglesforthisexteriorangle
Examinethetessellationdiagramabove,lookingforplaces
whereexterioranglesofatriangleoccur.(Again,youmayhave
toignoresomelinesegmentsandanglesinordertofocuson
trianglesandtheirverticalangles.)
Basedonseveralexamplesofexterioranglesoftrianglesinthediagram,writeaconjectureabout
exteriorangles.
Myconjecture:
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SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.5
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
ParallelLinesCutByaTransversal
Whenalineintersectstwoormoreotherlines,thelineiscalledatransversalline.Whentheother
linesareparalleltoeachother,somespecialanglerelationshipsareformed.Toidentifythese
relationships,wegivenamestoparticularpairsofanglesformedwhenlinesarecrossed(orcut)bya
transversal.Inthediagrambelow,∠1and∠5arecalledcorrespondingangles,∠3and∠6are
calledalternateinteriorangles,and∠3and∠5arecalledsamesideinteriorangles.
Examinethetessellationdiagramabove,lookingfor
placeswhereparallellinesarecrossedbyatransversal
line.
Basedonseveralexamplesofparallellinesand
transversalsinthediagram,writesomeconjectures
aboutcorrespondingangles,alternateinterioranglesand
samesideinteriorangles.
Myconjectures:
JustifyingOurConjectures
Inthenexttaskyouwillbeaskedtowriteaproofthatwillconvinceyouandothersthateachofthe
conjecturesyouwroteaboveisalwaystrue.Youwillbeabletouseideasabouttransformations,
linearpairs,congruenttrianglecriteria,etc.tosupportyourarguments.Agoodwaytostartisto
writedowneverythingyouknowaboutthediagram,andthenidentifywhichstatementsyoumight
usetomakeyourcase.Togetreadyforthenexttask,revisiteachoftheconjecturesyouwroteabout
andrecordsomeideasthatseemhelpfulinprovingthattheconjectureistrue.
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GEOMETRIC FIGURES – 5.5
Mathematics Vision Project
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READY Topic:PropertiesofQuadrilaterals
1. Usewhatyouknowabouttrianglestowriteaparagraphproofthatprovesthatthesumofthe
anglesinaquadrilateralis360º.
2. FindthemeasureofxinquadrilateralABGC.
Matchtheequationwiththecorrectlineinthegraphoflinesp,q,r,ands.
3. ! = !! ! + 2
4. ! = − !! ! + 2
5. ! = !! ! + 4
6. ! = − !! ! + 4
7. Describetheshapemadebytheintersection ofthe
4lines.Listasmanyobservationsasyou canaboutthe
shapeanditsfeatures.
READY, SET, GO! Name Period Date
r
pq
s
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Lesson 9
SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.5
Mathematics Vision Project
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SET Topic:Parallellinescutbytransversal,verticalanglesandexteriorangleofatriangleLabeleachpictureasshowingparallellineswithatransversal,verticalangles,oranexteriorangleofatriangle.Highlightthegeometricfeatureyouidentified.Canyoufindall3featuresin1picture?Where?
8. 9. 10.
11. 12. 13.
14. 15. 16.
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Lesson 9
SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.5
Mathematics Vision Project
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Findthevalueofthe2remoteinterioranglesinthefiguresbelow.
17. 18. 19.
Indicatewhethereachpairofanglesiscongruentorsupplementarybytrustinghowtheylook.Linespandqareparallel.
20. ∠5 !"# ∠8
21. ∠2 !"# ∠6
22. ∠2 !"# ∠8
23. ∠4 !"# ∠6
24. ∠3 !"# ∠5
25. ∠1 !"# ∠3
GO Topic:Complementaryandsupplementaryangles
Findthecomplementandthesupplementofthegivenangles.Itispossibleforthecomplementorsupplementnottoexist.
26. 37° 27. 59° 28. 89°
29. 111° 30. 3° 31. 90°
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Lesson 9
