1) Construct a segment [math]\overline{AB}[/math]. [br][br]2) Select the SLIDER [icon]/images/ggb/toolbar/mode_slider.png[/icon] tool. Select ANGLE. Set Min = [math]0^\circ[/math]. Set Max = [math]90^\circ[/math]. Set Increment = [math]1^\circ[/math]. [br] Hit OK. Slider should be named [math]\alpha[/math]. [br][br]3) Select the ANGLE WITH GIVEN SIZE [icon]/images/ggb/toolbar/mode_anglefixed.png[/icon] tool. [br] Select point [i]B[/i]. Then select point [i]A[/i] (vertex of soon-to-appear angle). [br] Then, where it says [math]45^\circ[/math], delete this and replace with [math]\alpha[/math] (see menu off to right). [br] Select COUNTERCLOCKWISE. If you do this correctly, you should see a new point [i]B'[/i] so that [br] [math]\angle BAB'=\alpha[/math]. [br][br]4) Construct a ray with endpoint B that passes through [i]B'[/i]. [br][br]5 )Select the ANGLE WITH GIVEN SIZE tool. Select point [i]A[/i]. Then select point [i]B[/i]. [br] Then, where it says [math]45^\circ[/math], delete this and replace with [math]\alpha[/math] (see menu off to right). [br] Select CLOCKWISE. You should see a new point A[i]'[/i] so that [math]\angle ABA'=\alpha[/math]. [br][br][color=#0000ff]More directions appear below the applet. [/color]
6) Construct a ray with endpoint [i]A[/i] that passes through [i]A'[/i]. [br][br]7) Plot the point of intersection of the 2 rays you've just constructed in steps (4) & (6). [br][br]8) Go to the algebra view (left). Hide the 2 rays, the original segment [math]\overline{AB}[/math], and the points [i]A'[/i] and [i]B'[/i]. [br][br]9) Use the POLYGON tool with vertices [i]A[/i], [i]B[/i], and the point of intersection of the 2 rays you constructed in step (7).
Display the length of all sides of this triangle. Move the slider around. What do you notice? What can we conclude about a triangle with 2 congruent angles?
[i]If two angles of a triangle are congruent, then the sides opposite those angles are also congruent[/i]. [br][br][b]Teachers:[/b][br]Here is an effective means for students to actively discover the Converse of the Isosceles Triangle Theorem for themselves.
What happens if we set [math]\alpha=60^\circ[/math]?
When this happens, all three angles measure [math]60^\circ[/math]. The resulting triangle is equiangular. And from our construction, we can now see the triangle is also equilateral. [br][br][b]Teachers:[/b][br]Here's one means for students to actively discover this corollary: [i]If a triangle is equiangular, then it is equilateral[/i].
[color=#0000ff]When you're done (or if you're unsure of something), feel free to check by watching the quick silent screencast below. [/color]