In figure, O is the centre of the circle, then which of the following is correct?

Consider the given figure,

Mark the angles as 1, 2, 3 and 4.

On the basis of given figure, we can conclude,

∠ 3 = ∠ 4

.
The angle at the centre is twice the angle at the circumference subtended by the same arc.
So,

∠ x = 2 ∠ 3

\Rightarrow ∠ x = ∠ 3 + ∠ 3

\Rightarrow ∠ x = ∠ 3 + ∠ 4

.....(i)

∠ y = ∠ 3 + ∠ 1

........(ii)

(i) - (ii)

gives,

\Rightarrow ∠ x - ∠ y = ∠ 4 - ∠ 1

.......(iii)
From triangle ACE,

\Rightarrow ∠ 4 = ∠ z + ∠ 1

\Rightarrow ∠ 4 - ∠ 1 = ∠ z

.........(iv)
From (iii) and (iv) we get,

\Rightarrow ∠ x - ∠ y = ∠ z

\Rightarrow ∠ z + ∠ y = ∠ x

∴ ∠ x = ∠ y + ∠ z

Hence, the correct relation is

∠ x = ∠ y + ∠ z

.
Therefore, option 1 is correct.

In figure, $O$ is the centre of the circle, then which of the following is correct?