In the figure,if $PA$ and $PB$ are tangents to the circle with center $O$ such that $\angle APB = 50^{\circ}$,then $\angle OAB$ is equal to: (in $^{\circ}$)

If a hexagon $ABCDEF$ circumscribes a circle,prove that $AB + CD + EF = BC + DE + FA$.

Write 'True' or 'False' and give reasons for your answer.
The tangent to the circumcircle of an isosceles triangle $ABC$ at $A$,in which $AB = AC$,is parallel to $BC$.

$A$ circle touches all the sides of a quadrilateral $ABCD$. If $AB = 8$,$BC = 10$,and $CD = 7$,then find the length of $AD$.

$A$ circle with centre $O$ touches sides $\overline{AB}$,$\overline{BC}$,$\overline{CD}$ and $\overline{DA}$ of quadrilateral $ABCD$ at points $P, Q, R$ and $S$ respectively. Prove that $m \angle AOB + m \angle COD = 180^{\circ}$ and $m \angle AOD + m \angle BOC = 180^{\circ}$.

In $\Delta ABC$,$\angle B$ is a right angle. If $AB = 8$ and $BC = 15$,find the radius of the incircle of $\Delta ABC$.