Question

In the figure, O is the centre of the circle, prove that $\angle x=\angle y+\angle z$.

Answer 1

Given : In circle, O is centre

To prove : $\angle x=\angle y+\angle z$

Proof : $\because \angle 3and\angle 4$ are in the same segment of the circle

$\therefore \angle 3=\angle 4..................\left(i\right)$

$\because$ Are AB subtends $\angle AOB$ at the centre and $\angle 3$ at the remaining part of the circle

$\therefore \angle x=2\angle 3=\angle 3+\angle 3$

$=\angle 3+\angle 4(\because \angle 3=\angle 4)....\left(ii\right)$

In $\Delta ACE$,

Ext. $\angle y=\angle 3+\angle 1$

(Ext. is equal to sum of its interior opposite angles)

$\Rightarrow \angle 3=\angle y-\angle 1...........\left(ii\right)$

From (i) and (ii),

$\angle x=\angle y-\angle +\angle 4..........\left(iii\right)$

Similarly in $\Delta ADF$,

Ext, $\angle 4=\angle 1+\angle z..........\left(iv\right)$

From (iii) and (iv)

$\angle x=\angle y-\angle 1+(\angle y+\angle z)$

Hence $\angle x=\angle y+\angle z$