Page 154 - IM_FL_Geometry_Print Sample

In this activity, students justify why the angle bisector is the set of points equidistant to the two rays that form the angle. This concept is essential for the next activity, where students reason that the three angle bisectors of a triangle meet at a single point that is equidistant to each side of the triangle.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5). Some points of interest haven’t been labeled for students so they will have to label their own points or search for other ways to communicate precisely (MP6).
Launch
If students have access to dynamic geometry software, suggest that it might be a helpful tool in this activity. Some points of interest haven’t been labeled for students. If necessary, suggest that labeling certain points may be helpful for explaining their ideas precisely.
If time allows, use dynamic geometry software to demonstrate that points on the angle bisector seem to be equidistant to the rays de ned by the angle, and that points that are equidistant to the rays seem to lie on the angle bisector. Ask students to explore along with you as you demonstrate. Create two rays and to form an angle . Create the angle bisector of angle and highlight it in a di erent color. Create a circle of arbitrary size and ask whether it’s possible to t the circle between the two rays so that they are tangent to the circle. Connect point to the center of the circle and ask students what they notice (the segment coincides with the angle bisector). Repeat this for several sizes of the circle to show that the center of the circle tangent to the two rays always seems to lie on the angle bisector. Ask students, “What can you say about the distance from the center of the circle to each ray?” Then ask, “What do you notice about the center of the circle?” Provide a similar demonstration for the idea that points on the angle bisector are always equidistant to the two rays. Choose a point on the angle bisector. Create a circle centered at this point and adjust the size until the circle is tangent to both rays at the same time. Repeat this by dragging the point to several locations along the angle bisector. Ask students what this means about points on the angle bisector in terms of their distance to the two rays. Doing this exploration gives students a chance to develop their conjecture that the angle bisector is the set of points equidistant to the two rays de ned by the angle more deeply than they did in the warm-up. However, it will take much more time to accomplish this.
Student Task Statement
Here is an angle .
Unit 7 Lesson 8: A Special Point 81