CIRCLES C9

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 ∠FHE and 
               ∠FGE in the same segment formed by 
                the chord EF (or arc EAF)

Required to prove: ∠FHE = ∠FGE

Construction : Join OE and OF.

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,

∠EOF = 2 ∠FHE ...............(i)

and,

∠EOF = 2∠FGE …………...(ii)


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

2∠FHE = 2∠FGE

2∠FHE = 2∠FGE                 (dividing by 2 both sides)
2               2

∠FHE = ∠FGE

∴ ∠FHE =∠FGE   proved!