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 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)

…… (1)

(Exterior angle is equal to the sum of two opposite interior angles) …… (2)

$\angle 4=\angle z+\angle 1$ (Exterior angle is equal to the sum of two opposite interior angles)

…… (3)

Adding (2) and (3)

……(4)

From equation (1) and (4) we have