In the figure,if $\angle OAB = 40^{\circ},$ then $\angle ACB$ is equal to: (in $^{\circ}$)

In a circle with centre $P$,$AB$ is a chord and point $C$ is a point other than $A$ and $B$ on the major arc $AB$. If $\angle ACB = 47^{\circ},$ then $\angle APB = $ .......... (in $^{\circ}$)

In the figure,$BC$ is a diameter of the circle and $\angle BAO = 60^{\circ}$. Then $\angle ADC$ is equal to: (in $^{\circ}$)

$AB$ is a chord of a circle with centre $P$. Point $C$ is a point other than $A$ and $B$ on the major arc $AB$. If $\angle ACB = 50^{\circ}$,then find $\angle APB$. (in $^{\circ}$)

In a circle with centre $P$,the length of a radius is $\sqrt{2} \ cm$. It is divided into two segments by chord $AB$ of length $2 \ cm$. If $C$ is a point on the major arc $AB$,then find the value of $\angle ACB$. (in $^{\circ}$)

Bisectors of angles $A, B$ and $C$ of a triangle $ABC$ intersect its circumcircle at $D, E$ and $F$ respectively. Prove that the angles of the triangle $DEF$ are $90^{\circ} - \frac{1}{2}A, 90^{\circ} - \frac{1}{2}B$ and $90^{\circ} - \frac{1}{2}C$.