Interior and Exterior Angles of Circles
Lesson 2 of 5
Objective: SWBAT prove relationships between the interior and exterior angles measures of circles and the measures of their intercepted arcs.
Activating Prior Knowledge
To begin this section, I give each student a copy of APK_Proving Circle Angle Relationships. I tell my students that they will have five minutes to work independently on the two problems. I also explain that the reasoning involved in solving the problems will become useful when we begin to write proofs in this lesson.
In any case, after the five minutes have passed, I will give students 2-3 minutes to consult with their seat partners in order to prepare for when I call non-volunteers to come and present their solutions.
After three minutes, I call random non-volunteers to the front of the class to present their answers to the two problems. As students are presenting, I am making sure that they are explaining and justifying their process precisely. For example, when a student says that the measure of angle B is 50 degrees, I coach them toward saying something more like "We know that angle B is an inscribed angle and therefore its measure is half that of its intercepted arc, which measures 100 degrees."
I am also careful to emphasize the inscribed angle theorem and the triangle exterior angle theorem, as these will play a prominent role in the proofs we write in this lesson.
To begin this section, I'll give each student a copy of Prove Angle Relations in Circles. I start by having students read the prompt for item 1, which says "Prove that the measure of each angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the arcs intercepted by the angle and its vertical counterpart."
At first, I approach this from a literacy-building standpoint. I have students identify the "two chords" referred to in the prompt. As students engage in pair shares with their seat partners, I confirm that the chords in question are segments BC and AD.
Next I call attention to the fact that the prompt makes mention of an "angle and its vertical counterpart". I explain that there are two such pairs of angles in the diagram (i.e., an angle and its vertical counterpart) I have each A-B pair take turns identifying an angle and its vertical counterpart. Then I identify these vertical pairs to make sure that all students are up to speed.
Next I explain that we could choose to focus on either of these pairs and that our choice of one pair or the other will be completely arbitrary. To drive home this point, I put it to a vote to decide which vertical pair we'll analyze. Supposing we pick angles CED and AEB, I would then have students do a quick pair share identifying the arcs that are intercepted by these angles.
Finally, I would guide students toward writing what we are trying to prove using precise math notation. I would first draw their attention to the phrase "the measure of each angle...is equal to half the sum of..." I would then write the following frame on the board:
"m<_______ = m<________ = 1/2(___________ + ____________)
Then I let students fill in the blanks. I walk around to see what students are writing and engage with students who are having trouble. I also direct students to check in with their seat partners and have discussion until they agree on how to fill in the blanks.
Supposing again that we've chosen to focus on angles CED and AEB, our next step will be to choose an auxiliary segment. See the following screencast to get an idea of how I get students to appreciate the process of strategically choosing an auxiliary segment.
Once students get the idea of choosing an auxiliary segment, it's time to prepare to write the proof. In the next screencast, I discuss how I get students to see and make use of the relationships that emerge after we've chosen a good auxiliary segment. Check it out.
Finally, I'll have students try their hands at writing the proof. I'll give them 5-10 minutes to work on it, and I allow them to consult with their partner as needed. When the time has elapsed, I'll show students the Completed Circle Interior Angle Theorem Proof and explain each of the steps.
In this section students will be writing a proof as they did in the previous section, but this time I will not do as much scaffolding. The proof in this section is highly analogous to the one from the previous section so this is a great opportunity to see what students learned from the previous proof and how well they can transfer their knowledge to this new context.
We'll be working on #2 on Prove Angle Relations in Circles. I'll begin by having the students determine what it is we're trying to prove. After a pair share, I'll randomly call on a non-volunteer to explain what we're proving and how we know.
After that, I'll be asking each student pair to list the possible auxiliary segments and to identify the one(s) that will actually suit our purposes. Then I'll randomly call on non-volunteers to share and explain their choices.
Next, I will instruct each pair of students to choose a DIFFERENT auxiliary line than their partner. This way, the partners will be working on similar, but not identical, proofs and they will appreciate the fact that we do have choices when we write proofs. Finally, it will be time for the students to work together, as needed, writing their respective proofs.
When students have had enough time to write their proofs and check them with their partner, I will randomly call on non-volunteers to bring their proofs under the document camera to present to the class. I leave the choice of two-column, paragraph, or flow proof up to students so I will try to get at least one sample of each format, if possible.