$O$ is centre of circle. $PB$ is tangent at $P$. If $\angle APB = x$, then find $\angle AOP$.

Because PB is a tangent at point P, angle APO is 90 degrees minus x. Since OA and OP are radii of the same circle, triangle AOP is an isosceles triangle, making angle OAP equal to angle APO. Therefore, angle OAP also equals 90 degrees minus x, and the sum of angles in triangle AOP means angle AOP equals 180 degrees minus 2 times (90 degrees minus x), which simplifies to 2x.

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