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