In the given figure O is the centre of the circle. Prove that: ∠x = ∠y + ∠z

The figure is given below:

AB is a chord of the circle. Chord AB makes an angle at the centre which is twice the angle it makes at a point on the circumference.

Hence w = v = x/2

In the quadrilateral, CGDH,

C = y (vertical angle)

the sum of angles in the quadrilateral CGDH is

180 - v + y + 180 - w + z = 360

Hence, w + v = x = y + z