In Fig. 16. 136, O is the centre of the circle, prove that ∠x=∠y+∠z.
We have,
∠3 = ∠4 (Angle on same segment)
By degree measure theorem,
∠x = 2 ∠3
∠x = ∠3 + ∠3
∠x = ∠3 + ∠4 (i) (Therefore, ∠3 = ∠4)
But,
∠y = ∠3 + ∠1 (By exterior angle property)
∠3 = ∠y - ∠1 (ii)
From (i) and (ii),
∠x = ∠y - ∠1 + ∠4
∠x = ∠y + ∠4 - ∠1
∠x = ∠y + ∠z + ∠1 - ∠1 (By exterior angle property)
∠x = ∠y + ∠z
Hence, proved