Question

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Answer 1

$\underline{\underline{\huge{\textbf{Solution:}}}}$

Given,
O is the centre of the circle

$\underline{\large{\textsf{Step-1:}}}$
Note angle subtended by arc XY and following result

Arc XY subtends $\angle{XOY}$ at Point 'O' and $\angle{XZY}$ at Point 'Z' on the the remaining part of the circle.

$\therefore \angle{XOY} = 2\angle{XZY}$ ————[1]

$\underline{\large{\textsf{Step-2:}}}$
Note angle subtended by arc XY and following result

Arc XY subtends $\angle{YOZ}$ at Point 'O' and $\angle{YXZ}$ at Point 'X' on the the remaining part of the circle.

$\therefore \angle{YOZ} = 2\angle{YXZ}$ ————[2]

$\underline{\large{\textsf{Step-3:}}}$
Adding [1] and [2]

$\implies \angle{XOY} + \angle {YOZ}= 2\angle{XZY} + 2\angle{YXZ}$

$\implies \angle{XOY} + \angle {YOZ}= 2(\angle{XZY} + \angle{YXZ})$ ————[3]

$\underline{\large{\textsf{Step-4:}}}$
Rewrite LHS of Equation [3]

We have,
$\angle{XOY} + \angle {YOZ}= \angle{XOZ}$ ————[4]

$\underline{\large{\textsf{Step-5:}}}$
Substitute [4] in [3]

$\therefore$
$\mathbf{\angle{XOZ} = 2(\angle{XZY} + \angle{YXZ})}$

$\textsf{Hence Proved}$