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In the figure, $O$ is the centre of the circle, prove that $\angle x = \angle y + \angle z$.
"
Given:
In the figure, $O$ is the centre of the circle.
To do:
We have to prove that $\angle x = \angle y + \angle z$.
Solution:
$\angle 4$ and $\angle 3$ are on the same segment
This implies,
$\angle 4=\angle 3$
$\angle x=2 \angle 3$ (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)
$\angle x=\angle 4+\angle 3$.............(i)
$\angle y=\angle 3+\angle 1$............(ii)
$\angle 4=\angle z+\angle 1$ (Exterior angle is equal to the sum of two opposite interior angles)
$\angle z=\angle 4-\angle 1$..............(iii)
Adding (ii) and (iii), we get,
$\angle y+\angle z=\angle 3+\angle 4$.............(iv)
From equations (i) and (iv), we get,
$\angle x=\angle y+\angle z$
Hence proved.
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