Prove that angles in the same segment of a circle are equal.

Given A circle C(O,r), having arc PQ and two angles ∠PRQ and ∠PSQ in the same segment of the circle

To prove ∠PRQ = ∠PSQ.

Construction Join OP and OQ.

Proof We know that the angle subtended by an arc at the centre is double the angle subtended by the arc at any point in the remaining part of the circle

We have, ∠POQ = 2∠PRQ and ∠POQ = 2∠PSQ

=> 2∠PRQ =2∠PSQ

=> ∠PRQ =∠PSQ