Write True or False and justify your answer in each of the following: Through three collinear points a circle can be drawn.

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 $^{\circ}$)

$ABCD$ is a quadrilateral such that $A$ is the centre of the circle passing through $B, C$ and $D$. Prove that $\angle CBD + \angle CDB = \frac{1}{2} \angle BAD$.

In cyclic quadrilateral $ABCD$,$\angle B = \angle D - 30^{\circ}$,then find $\angle B$. (in $^{\circ}$)

State whether each of the following statements is true or false:
$(1)$ The region between a chord and either of its arcs is called a sector.
$(2)$ The region between an arc and the two radii,joining the centre to the end points of the arc,is called a segment.

$AB$ 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 $\angle ACB + \angle APB = 135^{\circ}$,then find $\angle ACB$ and $\angle APB$.