Angles Inscribed in Circles

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Objective

SWBAT Prove the relationship between the measure of an angle inscribed in a circle and the measure of its intercepted arc.

Big Idea

Inscribed angles prove not to be half bad...and students prove that they are half the measure of their intercepted arcs.

Postulating Central Angle Measures

7 minutes

My goal in this section is to establish that the measure of a central angle is equal to the measure of its intercepted arc. Here's how I do it:

 

To put closure on this section I'll put the following sentence frame on the board:

Postulate: The measure of an angle whose vertex is _______________________ is equal to __________________________________________.

Then I'll have students work together in their A-B pairs to (1) determine what goes in the blanks and (2) rehearse reciting this postulate.

Inscribed Angle Measures

7 minutes

In this section, my goals are to (1) define inscribed angles, (2) distinguish inscribed angles from central angles, and (3) have students see and believe that the measure of an inscribed angle is half the measure of the intercepted arc.

I'll start by defining an inscribed angle as an angle with its vertex on the circle. I'll then ask the A partners in each A-B pair to sketch an example of an inscribed angle and then have the B partner say how this inscribed angle is different than a central angle.

On the shareout, I'd then make sure that we get around to saying something like:

Whereas the vertex of a central angle is at the center of the circle, the vertex of an inscribed angle is a point on the circle.

Once we've established this, I'll move on to goal 3 above. See the following screencast get a feel for how that goes.

 To put a cap on this section, I put the following sentence frame on the board:

Inscribed Angle Theorem: The measure of an angle whose vertex is _______________________ is equal to _______________________________________.

Then I'll have students work together in their A-B pairs to (1) determine what goes in the blanks and (2) rehearse reciting this theorem. Finally, we'll have some discussion on the implications of this being a theorem as opposed to a postulate. I.e., we'll be proving it...

Proving Case 1 of the Inscribed Angle Theorem

15 minutes

To fully prove the Inscribed Angle Theorem, we need to consider three distinct cases: 1) Center on the angle; (2) Center in the interior of angle; and (3) Center in the exterior of angle.

 

In this section I'll be guiding students through the reasoning for the proof of case 1. I'll hand out copies of Prove Inscribed Angle Theorem and then we'll get started. The process I of developing the proof is outlined in the following screencast:

 

Proving Case 2 of the Inscribed Angle Theorem

20 minutes

Now that we've proven case 1 of the theorem, I explain to students that we must exhaust all distinct cases of the theorem in order to truly prove it. I explain that the center of the circle must be either on, in, or outside of the angle. So these three cases are exhaustive.

I also explain that we will not need to prove anything that we've already proven for case 1. In other words, since we've proven case 1, the phrase "Proof of Case 1" can now be used as a reason in the proof of case 2. Students already have a copy of Prove Inscribed Angle Theorem and for this section we'll be working on page 2 of that document.

The most powerful step in this proof is the introduction of auxiliary segments. I don't expect all students to know which auxiliary segments to add so I model this for them and explain the rationale, as you can see in the following screencast:

When I have gotten students started with the auxiliary segments and the basic plan for the proof (for example, I tell them that they will need to use angle and arc addition postulates), I have them work together in pairs to write the proof.

After that, I select a few exemplars to share with the class.

Proving Case 3 of the Inscribed Angle Theorem

20 minutes

In this section students will use what they have learned thus far in the lesson to write their own proofs for case 3 of the inscribed angle theorem. I will have them work in pairs, but they will write the proof from start to finish without any modeling or guidance from me.

The diagram and blank space to write the proof are on page 3 of Prove Inscribed Angle Theorem, which students have already.

At the end of the allotted time, I'll collect these and see how individual students have done.