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Download Lesson 4.2 • Properties of Isosceles Triangles
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Lesson 4.2 • Properties of Isosceles Triangles In this activity you’ll learn how to construct an isosceles triangle (a triangle with at least two sides the same length). Then you’ll discover properties of isosceles triangles. Investigation 1: Base Angles in an Isosceles Triangle Sketch Step 1 In a new sketch, construct a circle with center A and control point B. Step 2 Construct radius AB. Step 3 Construct point C on the circle, and then construct radius AC. Step 4 !. Construct BC Step 5 Hide circle AB. A C B Investigate 1. Drag each vertex of your triangle. Explain why the triangle is always isosceles. 2. Measure !ACB and !ABC. Make sure the vertex is the second point you select. Angles ACB and ABC are the base angles of the isosceles triangle. Angle CAB is the vertex angle. Drag the vertices of your triangle and observe the measures of !ABC and !ACB. What do you observe about the measures? 3. Write a conjecture about the base angles of an isosceles triangle (Isosceles Triangle Conjecture). Investigation 2: Is the Converse True? In this sketch you will construct a triangle that has base angles of equal measure, then investigate the measures of the sides. Sketch Step 1 "!#. In a new sketch, construct AB Step 2 !# to form acute !CAB. Choose the Ray tool and construct AC Step 3 Mark !CAB as an angle of rotation by selecting, in order, points C, A, and B, then choosing Transform Mark Angle. Step 4 Double-click on point B to mark it as a center for rotation. C A B Steps 1 and 2 (continued) Discovering Geometry with The Geometer’s Sketchpad ©2008 Key Curriculum Press CHAPTER 4 55 Lesson 4.2 • Properties of Isosceles Triangles (continued) Step 5 "!# and choose Transform Rotate. Rotate AB "!# by the Select AB marked angle. Step 6 !#. Construct the point of intersection D of this line and AC D C A B Steps 5 and 6 Step 7 Hide the ray and the two lines, and leave all of the points showing. Step 8 Choose the Segment tool and construct "ADB. D C A B Steps 7 and 8 Investigate 1. Drag point C to change the measure of !CAB. What do you observe about "ADB? 2. Measure the sides of "ADB by selecting them and choosing Measure !! Length. Drag point C. What do you observe about the lengths of AD ! and BD ? 3. What do you observe about a triangle that has two angles of equal measure? Write your findings as a conjecture (Converse of the Isosceles Triangle Conjecture). EXPLORE MORE Use Sketchpad to investigate the statement “If a triangle is equiangular, then it is equilateral” and its converse, “If a triangle is equilateral, then it is equiangular.” Can you construct an equiangular triangle that is not equilateral? Can you construct an equilateral triangle that is not equiangular? Write a conjecture based on your investigation (Equilateral/Equiangular Triangle Conjecture). 56 CHAPTER 4 Discovering Geometry with The Geometer’s Sketchpad ©2008 Key Curriculum Press