Angles in the Same Segment of a Circle

Here’s a circle and two points P and Q on it. The points are joined to four arbitrary points A, B, C, and D on the circumference, lying on the same side of PQ.

Things to Explore

  1. Drag one of the points A, B, C, or D and observe the measures of the angles PAQ, PBQ, PCQ, and PDQ.
  2. Drag one of the points P or Q and observe the measures of the angles PAQ, PBQ, PCQ, and PDQ.

Questions

  1. As you drag A, B, C, or D, do the measures of the angles PAQ, PBQ, PCQ, or PDQ change?
  2. As you drag A, B, C, or D, do the angles PAQ, PBQ, PCQ, and PDQ remain equal to each other?
  3. As you drag P or Q, do the measures of the angles PAQ, PBQ, PCQ, or PDQ change?
  4. As you drag P or Q, do the the angles PAQ, PBQ, PCQ, and PDQ remain equal to each other?

That’s all for this exploration. Hope you’ve discovered some interesting properties of angles subtended in the same segment of a circle. Think about how you can formally prove what you discovered.

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