Chapter 4 Angles formed by 2 Lines being cut by a Transversal
6.4 Cut by a Transversal - MR. CONGLETON · 6.4 Cut by a Transversal A Solidify Understanding Task...
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SECONDARY MATH II // MODULE 6 SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.4 SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.4 Mathematics Vision Project mathematicsvisionproject.org 6. 4 Cut by a Transversal A Solidify Understanding Task Draw two intersecting transversals on a sheet of lined paper, as in the following diagram. Label the point of intersection of the transversals A. Select any two of the horizontal lines to form the third side of two different triangles. 1. What convinces you that the two triangles formed by the transversals and the horizontal lines are similar? 2. Label the vertices of the triangles. Write some proportionality statements about the sides of the triangles and then verify the proportionality statements by measuring the sides of the triangles. 3. Select a third horizontal line segment to form a third triangle that is similar to the other two. Write some additional proportionality statements and verify them with measurements. CC BY Lidyanne Aquino https://flic.kr/p/6zmScn Page 23 Horizontal lines are parallel o.BR F c Loc BELD by corresponding 1 S Door A E F Must be ENTIRE SIDE htt ftp.t.FIFAF.EE
Transcript of 6.4 Cut by a Transversal - MR. CONGLETON · 6.4 Cut by a Transversal A Solidify Understanding Task...
SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.4 SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.4
Mathematics Vision Project
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6. 4 Cut by a Transversal
A Solidify Understanding Task
Drawtwointersectingtransversalsonasheetoflinedpaper,asinthefollowingdiagram.LabelthepointofintersectionofthetransversalsA.Selectanytwoofthehorizontallinestoformthethirdsideoftwodifferenttriangles.
1. Whatconvincesyouthatthetwotrianglesformedbythetransversalsandthehorizontallinesaresimilar?
2. Labeltheverticesofthetriangles.Writesomeproportionalitystatementsaboutthesidesofthetrianglesandthenverifytheproportionalitystatementsbymeasuringthesidesofthetriangles.
3. Selectathirdhorizontallinesegmenttoformathirdtrianglethatissimilartotheothertwo.Writesomeadditionalproportionalitystatementsandverifythemwithmeasurements.
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Horizontallines are parallel
o.BR F cLocBELDby corresponding
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Door A E
F Mustbe ENTIRESIDE
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SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.4 SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.4
Mathematics Vision Project
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Tristanhaswrittenthisproportionforquestion3,basedonhisdiagrambelow:!"!" =!"!"
TiathinksTristan’sproportioniswrong,becausesomeofthesegmentsinhisproportionarenotsidesofatriangle.
4. CheckoutTristan’sideausingmeasurementsofthesegmentsinhisdiagramattheleft.
5. Nowcheckoutthissameideausingproportionsofsegmentsfromyourowndiagram.Testatleasttwodifferentproportions,includingsegmentsthatdonothaveAasoneoftheirendpoints.
6. Basedonyourexamples,doyouthinkTristanorTiaiscorrect?
Tiastillisn’tconvinced,sinceTristanisbasinghisworkonasinglediagram.Shedecidestostart
withaproportionsheknowsistrue:!"!" =!"!".(Whyisthistrue?)
Tiarealizesthatshecanrewritethisproportionas!"!!"!" = !"!!"!" (Whyisthistrue?)
CanyouuseTia’sproportiontoprovealgebraicallythatTristaniscorrect?
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Still proportional
6.4RSG
SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.4
Mathematics Vision Project
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READY Topic:Pythagoreantheoremandproportionsinsimilartriangles.Findthemissingsideineachrighttriangle
1. 2.
3. 4.
Createaproportionforeachsetofsimilartriangles.Thensolvetheproportion.
5. 6.
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READY, SET, GO! Name Period Date
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SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.4
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SETTopic:ProportionalityoftransversalsacrossparallellinesForquestions7and8,writethreeequalratios.
7. Thelettersa,b,canddrepresentlengthsoflinesegments.
8.
9. Writeandsolveaproportionthatwillprovidethemissinglength.
10. Writeandsolveaproportionthatwillprovidethemissinglength.
Forquestions11–14findandlabeltheparallellines.(i.e.!" ∥ !")Thenwriteasimilaritystatementforthetrianglesthataresimilar.(i.e.∆ !"# ~ ∆ !"#)11. 12.
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SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.4
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13. 14.
GO Topic:SimilarityinslopetrianglesEachlinebelowhasseveraltrianglesthatcanbeusedtodeterminetheslope.Drawinthreeslope-definingtrianglesofdifferentsizesforeachlineandthencreatetheratioofrisetorunforeach.15. 16.
17. 18.
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241236 42
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SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
6. 5 Measured Reasoning
A Practice Understanding Task
Findthemeasuresofallmissingsidesandanglesbyusinggeometricreasoning,notrulersandprotractors.Ifyouthinkameasurementisimpossibletofind,identifywhatinformationyouaremissing.Linesp,q,r,andsareallparallel.
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SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1. Identifyatleastthreedifferentquadrilateralsinthediagram.Findthesumoftheinterioranglesforeachquadrilateral.Makeaconjectureaboutthesumoftheinterioranglesofaquadrilateral.
Conjecture:
2. Identifyatleastthreedifferentpentagonsinthediagram.(Hint:Thepentagonsdonotneedtobeconvex.)Findthesumoftheinterioranglesforeachpentagon.Makeaconjectureaboutthesumoftheinterioranglesofapentagon.
Conjecture:
3. Doyouseeapatterninthesumoftheanglesofapolygonasthenumberofsidesincreases?Howcanyoudescribethispatternsymbolically?
4. Howcanyouconvinceyourselfthatthispatternholdsforalln-gons?
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A gon 180in 360
Teacher saidno
SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.5
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READY Topic:PythagoreanTheoremandratiosofsimilartrianglesFindthemissingsideineachrighttriangle.Trianglesarenotdrawntoscale.
7. Basedonratiosbetweensidelengths,whichoftherighttrianglesabovearemathematicallysimilartoeachother?Providethelettersofthetrianglesandtheratios.
READY, SET, GO! Name Period Date
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SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.5
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SET Topic:Usingparallellines,andanglerelationshipstofindmissingvalues.
Ineachofthediagramsusethegiveninformationprovidedtofindthemissinglengthsandanglemeasurements.
8. Linem∥nando∥p,findthevaluesofanglesx,yandz.Also,findthelengthsofa,bandc.
9. Lineq ∥r∥sandt∥uandp∥w ∥v,findthevaluesofanglesx,yandz.Also,findthelengthsofa,b,c,d,e,f.
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SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.5
Mathematics Vision Project
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GO Topic:Solveequationsincludingthoseincludingproportions
Solveeachequationbelow.10. 11. 12.
3! − 5 = 2! + 7 57 =
!21
3! =
185! + 2
13. 14. 15.
12 ! − 7 =
34 ! − 8
17 + 3(! − 5) = 2(! + 3) ! + 56 = 3(! + 2)
9
16. 17. 18.
! + 2 + 3! − 8 = 90 512 =
!8
45 =
! + 215
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