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* *Reflection: Rigor
Angles Inscribed in Circles - Section 3: Proving Case 1 of the Inscribed Angle Theorem

Before the CCSS I had taught the proof of the inscribed angle theorem, but I had only taught students how to prove the most convenient case: the one where the side of the angle goes through the center.

With the focus on math practices that came with the CCSS, I now feel compelled to teach the proof of all three cases. Not because I feel all three cases are so important but because it is a more faithful representation of what mathematicians actually do and how mathematics is as a discipline. To not include all the cases is to not be rigorous. It is to leave room for there to be cases in which what we are trying to prove is not true. Thinking in this way, we have to reason about all possible cases and try to put them into categories that are exhaustive. In this way, we're teaching more than the content, we're teaching the discipline. That was an 'Aha!' moment for me in this lesson.

*Mathematical Practices not on the list*

*Rigor: Mathematical Practices not on the list*

# Angles Inscribed in Circles

Lesson 1 of 5

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

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.

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#### Inscribed Angle Measures

*7 min*

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...

*expand content*

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:

*expand content*

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.

*expand content*

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.

*expand content*

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- UNIT 1: Community Building, Norms, and Expectations
- UNIT 2: Geometry Foundations
- UNIT 3: Developing Logic and Proof
- UNIT 4: Defining Transformations
- UNIT 5: Quadrilaterals
- UNIT 6: Similarity
- UNIT 7: Right Triangles and Trigonometry
- UNIT 8: Circles
- UNIT 9: Analytic Geometry
- UNIT 10: Areas of Plane Figures
- UNIT 11: Measurement and Dimension
- UNIT 12: Unit Circle Trigonmetry
- UNIT 13: Extras