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Chap 3-Triangles.Pdf Defining Triangles Name(s): In this lesson, you'll experiment with an ordinary triangle and with special triangles that were constructed with constraints. The constraints limit what you can change in the triangie when you drag so that certain reiationships among angles and sides always hold. By observing these relationships, you will classify the triangles. 1. Open the sketch Classify Triangles.gsp. BD 2. Drag different vertices of h--.- \ each of the four triangles / \ t\ to observe and compare AL___\c yle how the triangles behave. QI Which of the kiangles LIqA seems the most flexible? ------------__. \ ,/ \ Explain. * M\----"--: "<-----\U \ a2 \Alhich of the triangles seems the least flexible? Explain. -: measure each I - :-f le, select three l? 3. Measure the three angles in AABC. coints, with the rei€x I loUr middle I 4. Draga vertex of LABC and observe the changing angle measures. To Then, in =ection. I le r/easure menu, answer the following questions, note how each angle can change from I :hoose Angle. I being acute to being right or obtuse. Q3 How many acute angles can a triangle have? ;;,i:?:PJi;il: I qo How many obtuse angles can a triangle have? to classify -sed I rc:s. These terms f> Qg LABC can be an obtuse triangle or an acute :"a- arso be used to I : assify triangles I triangle. One other triangle in the sketch can according to I also be either acute or obtuse. Which triangle is it? their angles. I aG Which triangle is always a right triangle, no matter what you drag? 07 Which triangle is always an equiangular triangle? -: -Fisure a tensth, I =ect a seoment. I thenlin t'e p t,'leasure.. 5. Measure the lengths of the three sides of LABC. menu, I :-coselength.l z n t t a An- 1 r i, r . 6. Drag a vertex of LABC and observe the changing side lengths. furonng Geometry with The Geometer's Sketchpad Chapter 3: Triangles . 63 tory2 Key Curriculum Press { Defining Triangles (continued) Scalene. isosceles, l-9 none the side lengths are equal, the triangle and equilateral are aB If of I terms used to I is a scnlene triangle. If two or more of the sides classify triangles by I are equal in length, the triangle is isosceles.lf all relationships among I their sides. Because I three sides are equal in length, itis equilaternl. AABC has no I constraints, it can be I Which type of triangle is AABC most of the time? any type of triangle. I Q9 Name a triangle besides LABC that is scalene most of the time. (Measure if you like, but if you drag things around, you should be able to tell without measuring, just by looking.) Qro Which triangle (or triangles) is (are) always isosceles? a{r Which triangle is always equilateral? Q12 For items a-e below, state whether the triangle described is possible or not possible. To check whether each is possible, try manipulating the triangles in the sketch to make one that fits the description. If it's possible to make the triangle, sketch an example on a piece of paper. a. An obtuse isosceles triangie b. An acute right triangle c. An obtuse equiangular triangle d. An isosceles right triangle e. An acute scalene triangle Q{3 Write a definition for each of these six terms: scute triangle, obtuse triangle, right trinngle, scnlene triangle, isosceles triangle, and equilateral trinngle. Use a separate sheet if necessary. ln the Display menu, choose show Ail Hidden EXplOre MOre to get an idea of I how the triangles in the Glassify 1. Open a new sketch and see if you can come up with ways to construct Triangles.gsp a right triangle, an isosceles triangle, and an equilateral triangle. sketch were constructed. Describe your methods. 64 . Chapter 3: Triangles Exploring Geometry with The Geometer's Sketchpad @ 2002 Key Curriculum Press friangte Sum Name(s): conjectureabout":::'**'i?:i':::.1*'rrinvestiga*FT:^1-: two-part invesrigation. First you,rr";;t;;,k"ff - To measure l"T;jliffi an J :ill:,#*trq*,fl sketch and t*T,J:",n"" ;1?',3;:frilfrii rnvestisate *,fi3j*,J5ifu/ 1 Construct AAB]. choose anere l-t 2' M"usure its ChooseCalculate three q! (6rsD. I 'uuse sdlcUlate - -- angres. itx:E::i::t tz-/tEi(j / = 87.2 Calculate mZBcA= 4e'5o t{ff,$idffi f : the sum of the angle measures. once on u I +. Draga vertex *r' / *of tlthe triangte and n*== '.".,,".1nitJunr"iii''io i il:;;::::the angle sum. observe calcutation, 1 J _ seiecr ai What is the sum ooint B I orF the angles in any Follow fuianele? ,"?#43#:i/ these steps to investigate why,J'ff;:l";; s. Consrructr.-orsrruct a tine through point " .;;";:":ilf;:;:: I : B paraltelparalret to At. " 6' Construct the midpoints cnoose of A-B and CE. "f"j,"t.lii*il,:Triangls l=i _ Lii.i.',-'.i 7. Consfruct F the interio r of AABC. enter ror ,?t?,*jffi ro ta ti on and rota lransform ?ii,ffi te the men,, I[',Hi'?81'r,f""#ii ;c choose Rotate' 9' 1 Give the new fuiangle interior a different B:Fil'#'lf ( color. 'o' as a center and rotate Y:* ffi"J:,T,:idpoint the interior by 180" 11. Give this new tuiangle interior a different color. B 72. the three triangres 3,ff ilffit f,:f,|l',iffiff* are rerated to *:t:,lixftI;j*:ll#;;r?f each u--' '"vtd'ut now j#:",*d,..ne.r,he conjecfure from e1. i:f ,Tl,this demonsfrates your Chapter ar tri"nge, .ii Exterior Angles in a Triangle Name(s): Anexterior angle of a triangle is formed when one of the sides is extended. An exterior angle lies outside the triangle. In this investigation, you'll discover a relationship between an exterior angle'and the sum of the measures of the two remote interior angles Sketch and lnvestigate 1. Construct AABC. nICAB = 45.5o Hold down the nZABC = 67.2" mouse button on the 2. Construct AC--) to extend side AC. Segment tool and drag right to choose mtBCD=112.74 the Ray tool. 3. Construct point D onA-, outside of the triangle. Select, in order, A C D' points B, C, and D. Measure exterior angle BCD. Then, in the Measure menu, Measure the remote interior angles ZABC and ICAB. choose Angle. 6. Drag parts of the triangle and look for a relationship between the measures of the remote interior angles and the exterior angle. Choose Galculate I ["#']H$%Tiil' f'qr How are the measures of the remote interior angles related to the Calculator. Click I measure of the exterior angle? Use the Calculator to create an once on a I measurement to I expression that confirms your conjechrre. enter it into a I calculation. I I rt I Select the interior; ( marked vector. il then, in the I Transform menu, | 10. Give the new triangle interior choose Translate. I a different color. 11,. Construct the midpoint of BC Double-clickthelrrn as a and rotate the triangle point to mark it as a( tz' Mark the midpoint center for rotation center. Select the I interior about this point by 180". interior; then, in the I Transfrom menu, I n. choose Rotate. IJ. Give this new triangle interior a different color. a2 Explain how the two angles that fill the exterior angle are related to the remote interior angles in the triangle. Explain how this demonstrates your conjecture from Q1. Use the back of this sheet. 56 . Chapter 3: Triangles Exploring Geometry with The Geometer's Sketchpad @ 2OO2 Key Curriculum Press Triangle lnequalities Name(s): In this investigation, you'll discover relationships among the measures of t the sides and angles in a triangle. Sketch and lnvestigate 1. Construct a triangle. -: 'reasure a side, m AB- = 2.7 cm r=.ct it; then, in the mBe=3.6cm lr4easure menu, 2. Measure the lengths of . rloose Length. the three sides. mffi=3.4cm - -:ose Calculate '':rn the Measure 3. Calculate the sum of any m AB-+m BT = 6.31 cm -enu to open the two side lengths. lalculator. Click once on a 'neasurement to 4. Drag a vertex of the enter it into a calculation. triangle to try to make the sum you calculated equal to the length of the third side. Ql Is it possible for the sum of two side lengths in a triangle to be equal to the third side length? Explain. Q2 Do you think it's possible for the sum of the lengths of any two sides of a triangle to be less than the length of the third side? Explain. t Q3 Summarize your findings as a conjecture about the sum of the lengths of any two sides of a triangle. To measure an [, 5. Measure IABC, ICBA, and IACB. =-ole, select three points, with the .:1ex your middle 6. Drag the vertices of your triangle and look for relationships between =:lection. Then, in --: Measure menu, side lengths and angle measures. I choose Angle. Q4 In each area of the chart below, a longest or shortest side is given. Fill in the chart with the name of the angle with the greatest or least measure, given that longest or shortest side. AB longesl Largest AC longesl Largest BC longesl Largest side angle? side angle? side angle? AB shortest Smallest AC lo.tgest Smallest BZlongest Smallest side angle? side angle? side angle? Q5 Summarize your findings from the chart as a conjecture. ! :rnloring Geometry with The Geometer's Sketchpad Chapter 3: Triangles .
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