Tuesday, 25 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 ∠USV and ∠UTV in the same segment formed by the chord UV (or arc UAV)
Required to prove: ∠USV = ∠UTV
Construction : Join OU and OV.
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,
∠UOV = 2 ∠USV ...............(i)
∠UOV = 2∠UTV …………...(ii)
Since (i) = (ii), we get,
2∠USV = 2∠UTV
2∠USV = 2∠UTV (dividing by 2 both sides)
∠USV = ∠UTV
∴ ∠USV =∠UTV proved!