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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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Updated on: 10-Oct-2022

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