Ms. Saleh's tenth-grade geometry class is working on an inscribed triangle problem. Students had previously worked with angles, triangles, and circles and had been introduced to the idea of a proof.

Ms. Selah's instructions: "Work in your groups to construct an inscribed triangle in the top of a semi-circle. Here's an example: You'll need to use your compass, protractor and straightedge to do so, and you may choose any random point C on the semi-circle, you don't have to make the triangle look like the example. After you have your triangle, measure angles A, B, and C for at least five different cases and record your findings in a table. I'm particularly interested in what you find out about angle C. In a few minutes I'll come around to each group to hear your observations."

One group of students created this chart.

Here is a dialogue between Ms. Saleh and members of one group:

Ms. Saleh: Can you describe how you worked on the problem?

Ben: We made a circle with the compass, and then drew lines.

Ms. Saleh: How did you draw the lines?

Ben: We found a place on the top of the circle and drew lines to either corner.

Sonia: See these are our points. {she points to the top of the circle] we actually tried a bunch of different points for C.

Ms. Saleh: What about that point in the middle, O?

Grant: We thought we'd need that because last time we did circles we used it. But we didn't use it to draw with.

Ms. Saleh: Okay, we may want to come back to it later, so it wasn't a mistake to put it in. Now tell me about your chart. Can you draw any conclusions about angle C?

Grant: Well, we measured it in five positions on the circle. It is always close to 90 degrees, although it's never exact.

Ben: No, it was exact once.

Grant: Oh, I see. Yes, but it was just once.

Ms. Saleh: Do you have any ideas about angle C?

Sonia: Yes, it's always the top of the triangle and C always looks like a right triangle. So I think C is a right triangle, no matter what.

Ms. Saleh: Yes, it certainly looks that way. But does the table support the idea that angle C will be a right angle in every case? We have some data, 94 degrees for instance, that isn't a right angle.

Grant: That's just cuz we measured with a protractor and maybe it was off a little.

Ms. Saleh: If we had a more accurate protractor and many more triangles how could we show that angle C is always a right triangle? How many measurements would it take?

Ben: You'd have to do every single triangle ever. And that's not possible, right?

Ms. Saleh: Sonia what do you think? Could we check every possible angle C?

Sonia: No, I couldn't at least. Maybe I could make an argument about why it needs to be 90 degrees.

Ms. Saleh: That's great. Please work on an argument you think will convince one another that angle C is always a right angle. Then I'll come back and you can try to convince me!

  Reflect on the Inscribed Triangle problem