In the given figure, ABCD is a cyclic quadrilateral, OB is the radius, PB is the tangent at point B and $∠ O B C = 30^{\circ}$ . AOC is a straight line. Then, find the value of x.

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Solution

In the given figrue,
$∠ O B P = 90^{\circ}$
and $∠ O B C = 30^{\circ}$

$∴ ∠ C B P = 90^{\circ} - 30^{\circ} = 60^{\circ} \Rightarrow ∠ C A B = 60^{\circ}$
$[∵ ∠ C A B = ∠ C B P$ , alternate segment of chord]
In $Δ O A B,$
OA =OB
$\Rightarrow ∠ O A B = ∠ O B A = 60^{\circ}$
In $Δ O A B,$
$∠ O A B + ∠ O B A + ∠ A O B = 180^{\circ} \Rightarrow 60^{\circ} + 60^{\circ} + ∠ A O B = 180^{\circ} ∴ ∠ A O B = 180^{\circ} - 120^{\circ} = 60^{\circ}$

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In the given figure, ABCD is a cyclic quadrilateral, OB is the radius, PB is the tangent at point B and ∠OBC=30∘. AOC is a straight line. Then, find the value of x.

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In the given figure, ABCD is a cyclic quadrilateral, OB is the radius, PB is the tangent at point B and $∠ O B C = 30^{\circ}$ . AOC is a straight line. Then, find the value of x.