Tuesday, 11 February 2014
In the following figure, prove that the angles in the same segment of a circle are equal.
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.
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)
∠POQ = 2∠PSQ …………...(ii)
Since (i) = (ii), we get,
2∠PRQ = 2∠PSQ
2∠PRQ = 2∠PSQ (dividing by 2 both sides)
∠PRQ = ∠PSQ
∴ ∠PRQ =∠PSQ proved!