Question

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

Answer 1

It is given that, O is the center of circle and A, B and C are points on circumference on triangle

We have to prove that ∠x = ∠y + ∠z

∠4 and ∠3 are on same segment

So, ∠4 = ∠3           
    `angle x = 2 angle3`  (Angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of the circle)

 `anglex = angle 4 + angle 3`               …… (1)

 `angley= angle 3 + angle 1`           (Exterior angle is equal to the sum of two opposite interior angles)         …… (2)

\[\angle4 = \angle z + \angle1\]

   (Exterior angle is equal to the sum of two opposite interior angles)  

  `anglez= angle 4 -  angle 1`              …… (3)

Adding (2) and (3)

  `angley + angle z  = angle 3 + angle 4`     ……(4)

From equation (1) and (4) we have 

 `anglex = angle y + angle z`