Thoughts About Inscribed Angles and Arcs

In the past year and a half or so, I've been assigned to team teach with several different math teachers. I've been assigned to different schools in Brooklyn, with different pacing calendars. Because of this, I have happened to be in Geometry classes with three other teachers as they covered the various topics relating to circles: diameters, radii, chords, tangent, secants, arcs and angles. This past week, something occurred to me as I watched and assisted in another lesson.

When presenting the initial definition of inscribed angles and stating the relationship between the inscribed angle, the central angle and the intercepted arc, I generally see the same initial image.

This is a fine place to start, with a dart-shaped object inside the circle, somewhat similar in form to the insignia of Star Fleet. (Well, that's why I like it.) But teachers have to make sure that they address possible misconceptions.

• Note that angle AOB is formed by two radii, which by definition are congruent. Angle ACB is formed by two chords? Is chord AC congruent to BC? They look that like or at least look close enough that the student might believe that the teacher meant to draw them that way. In truth, they might be congruent, but they don't have to be.
  
  
• Does the vertex of the inscribed angle have to be somewhere "behind" the center of the circle? The problem with the dart-like figure is that the center of the circle is situated between the two chords. This doesn't have to be true. Inscribed angles can have their vertex anywhere on the circle, with the exception of inside the intercepted arc being considered. If can be to the side so that the chords intersect the radii. It might be closer to the intercepted arc than the center of the circle. Points like this should be shown, although, admittedly, some points would be difficult to illustrate.
  
  
  
  
  
• Another point rarely made: after covering that the central angle equals the intercepted arc and that the inscribed angle is half the size of the intercepted arc, the next logical step doesn't get taken. Compare multiple inscribed angles intercepting the same arc. What is their relationship? Why?
  
  
  
  
  
  

Finally, there's a Special Case, which doesn't have to be presented as such -- at least, not the first time they see it. I see the special case just given that if an inscribed angle intercepts a semicircle, usually marked off by a diameter, then the inscribed angle is ________. STOP! Don't tell them. Ask them to figure it out using the rule for inscribed angles. Remind them, if necessary, that a diameter is also a straight angle with the center of the circle as its vertex. What is the measure of a straight angle?

What kind of angle does it have to be? Will it always be that for every inscribed angle intercepting a semicircle? Why or why not? What kind of triangle is inscribed in the circle? What do we know about the other two angles (taken together)?

With a little more variation in the foundation of the material, students will be better prepared for more complicated problems with "busier" images with criss-crossing chords or inscribed triangles and quadrilaterals.