In figure, O is the centre of the circle, then prove that ∠x = ∠y + ∠z.

We have,

∠3 = ∠4 [Angles in same segment]

∴ ∠x = 2∠3 [By degree measure theorem]

⇒ ∠x = ∠3 + ∠3

⇒ ∠x = ∠3 + ∠4 ... (i) [∠3 = angle 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