The goal of this discussion is for students to understand the Inscribed Angle Theorem.

Ask students to share their conjectures. Press them to attend to precision in the language they use. Next, display these images for all to see:

Ask students to compare and contrast these images. (All three images have a central angle and an inscribed angle . In the first case the central angle is less than 180 degrees, in the second it’s exactly 180 degrees, and in the third it’s more than 180 degrees. In all three cases, the inscribed angle is half the measure of the central angle, so this supports our conjecture.) Do the ideas support or contradict the class conjectures?

Add the following assertion to the class reference chart, and ask students to add it to their reference charts. Tell students that this can be proven to be true, but since we haven’t done so in class, we will add it as an assertion:

Inscribed Angle Theorem: The measure of an inscribed angle is half the measure of the central angle that defines the same arc. (Assertion)

Students may assume that  is the center of the circle. Ask these students to mark a point that appears to be the center of the circle and label it .

If students struggle to connect the arc measurements to the inscribed angle measurements, suggest they mark the center of the circle and then draw a central angle using radii that intersect the circle at points and . Then suggest they look at their work from the previous activity and try to find connections.

The goal of this discussion is to generalize to all pairs of intersecting chords.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to to prove that triangles and are similar. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner’s ideas and writing.

If time allows, display these prompts for feedback:  

• “Your explanation tells me .”
• “Can you say more about ?”
• “A detail (or word) you could add is , because .”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

• “How many pairs of angles do you need to show are congruent in order to show the triangles are similar?” (We need to show that two pairs are congruent. The third pair of angles must then be congruent because the angle measures in a triangle always add to 180 degrees.)
• “What if we didn’t have the measures of the arcs, but we just had the diagram? Could we still prove the triangles are similar?” (Yes. Angles and are congruent no matter what the measures of the arcs are because they are vertical angles. And angles and are inscribed in the same arc, so they will always be the same measure—half of that arc.)