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.

(blog): 360, 180, 90, 2 and 1/2: I'm Talking Arcs and Inscribed Angles

Assisting today in a Geometry class. It was Day 2 of Arcs of a Circle. Yesterday, the teacher covered central angles with the class, which meant that today's lesson moved to the inscribed angles. She carefully and diligently explained what makes an inscribed angle, and how the line segments intercept an arc in a similar way that they saw in the previous lesson. And then we got into the relationship between the arcs and the two types of angles.

The measure of an arc of a circle is equal to the size of the central angle which intercepts it. The measure of the inscribed angle is half the size of the arc it intercepts. Pause. What does this make the relationship between the central and the inscribed angle. Pause. Wait. Rephrase? Response? Good -- but try again in a full sentence...

"It's half the size." sounds good. No, really, it does -- it means someone's paying attention and either getting it, or somewhat getting it. Following up: "What is half of what?" The inscribed angle is half of the central angle.

Okay. So if the inscribed angle is 60 degrees, how big is the central angle? Let them think about it. Did they come up with 120 degrees? Or 30 degrees? If the smaller angle is half the bigger angle, then the bigger angle is ... ? (Okay, it's a leading question, and I hate leading questions, but sometimes you do need to just pull that one number out of them so you can move on.)

They moved along with the notes and did a couple of practice problems before moving on to the next step. What if two inscribed angles intercepted the same arc? What could we conclude about the two angles? The teacher waited to see if they could reach the statement before she gave it. It took a moment to realize that A was half the arc and B was half the same arc, so angles A and B had to be congruent even if we didn't know how big the arc was. We didn't need to know. But if we did, we could work things out.

Then things started getting complicated because when you start putting in two many line segments and too many inscribed angles, triangles start forming. Wait! What are we supposed to do with those?! Treat them like three inscribed angles, of course, but don't forgot those properties of triangles, either. Particularly, the one about the sum of the angles!

So if we had a problem that looked like this:

... we have enough information to fill in both angles BAC and BCA as well as arcs AC and BC. We just might not know that we know yet. Not unless we remember some other facts about circles and triangles. The total measure of the central angles in a circle is 360 degrees, so the total of the arcs of the circle is also 360 degrees. The sum of the angles of a triangle is 180 degrees. Notice that if each angle of the triangle is inscribed that make each part of the circle twice the size of inscribed angle -- and 360 degrees is twice as big as 180!

Excellent discovery, if they can make it on there own. One student was hovering about it while he was talking. If he made the connection, he didn't share it with me, but he was close to it.

Finally, why is 90 so important that I included it in the title?

Because many of these problems use diameters as one of the line segments. A diameter cuts the circle in half, into two semicircles, each 180 degrees. The central angle formed by the two radii joining into a diameter is a straight angle, measuring 180 degrees. Any inscribed triangle using the diameter as one of its sides would, by necessity, have an angle that measures half of 180 degrees, which is 90 degrees.

Wait a minute!

So any inscribed triangle using the diameter of a circle is a right triangle? And any inscribed right triangle has to include the diameter?

It's almost as if someone planned it that way. Maybe not, but that's how we planned the lesson.