Inscribed Angle Theorem


Activity #1

Recall that in a circle, an INSCRIBED ANGLE is an angle whose vertex lies on the circle, and its sides are chords that interact the circle at two distinct points. In the activity below, the blue angle is an INSCRIBED ANGLE that intercepts the red arc. The red angle is a central angle (of the circle) that also intercepts the same red arc. In fact, this applet was designed so that both the inscribed angle and central angle always intercept the same red arc. 1) Drag the green slider all the way to the right in the applet below and watch what happens. 2) Now drag the green slider all the way back. Move any one or more of the blue and/or red points around and repeat step (1).

Answer each one of the questions below so they can be submitted in the final activity. Use either the PDf or a piece of paper. Answer in complete sentences: 1) How does the measure of any central angle of a circle compare with the measure of its intercepted arc? 2) According to what you've observed in the activity above, how does the measure of the inscribed angle compare with the measure of the central angle (that intercepts the same arc?) 3) Use your results from (1) and (2) to describe how one could find the measure of an inscribed angle given the measure of the arc it intercepts.

Activity 2. Inscribed Angle meets the Diameter

In this activity, the central angle always remains a straight angle (180 degrees). Therefore the intercepted arc is a semicircle. Click on the pink checkbox to show the inscribed angle. Notice how the inscribed angle and central angle both intercept the same arc. Use the inscribed angle theorem (activity 1) you've just learned to make a conjecture, an educated guess, as to what the measure of the inscribed angle in this applet should be. Be sure to move points B, C, and the pink vertex of the inscribed angle around as well. (You can also change the radius of the circle if you wish.)