Question

in the given figure , o is the centre of the circle .prove that angle x = angle y + angle z .

Answer 1

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

   hence  w = v = x/2

in the quadrilateral,  CGDH,
         angle C = y  (vertical angle)

 sum of angles in the quadrilateral CGDH is

     180 - v + y + 180 - w + z = 360

  hence,         w + v  =    x = y + z

Answer 2

its your answer, Thanks for the question.