CIRCLES C1

In the following figure, prove that the angles in the same segment of a circle are equal.

Solution

Given : A circle with centre O and the angles ∠PRQ and 

               ∠PSQ in the same segment formed by 

                the chord PQ (or arc PAQ)

Required to prove: ∠PRQ = ∠PSQ

Construction : Join OP and OQ.

Proof :

Try to remember that angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle, hence we have,

∠POQ = 2 ∠PRQ ...............(i)

and,

∠POQ = 2∠PSQ …………...(ii)

Since (i) = (ii), we get,

2∠PRQ = 2∠PSQ

2∠PRQ = 2∠PSQ                 (dividing by 2 both sides)

2               2

∠PRQ = ∠PSQ

∴ ∠PRQ =∠PSQ   proved!