Angles Subtended By Circular Arcs

The simulation shows a circle and the angles subtended by its arc PQ at its center and circumference.

Things to Explore

1. Drag the point R and observe the measures of ∠POQ and ∠PRQ.
2. Drag the points P and Q and observe the measures of ∠POQ and ∠PRQ.

Questions

1. Is there some relation between the measures of ∠PRQ and ∠POQ?
2. If you drag R, does the measure of ∠PRQ change?
3. If you drag R, does the relation between the measures of ∠PRQ and ∠POQ change?
4. If you drag P or Q, do the measures of ∠PRQ and ∠POQ change?
5. If you drag P or Q, does the relation between the measures of ∠PRQ and ∠POQ change?

That’s all for this exploration. Hope you’ve discovered some interesting properties of angles subtended by an arc of a circle at its center and circumference. Think about how you can formally prove what you discovered.