In a cyclic quadrilateral,$\angle A = 3x + 15^{\circ}$ and $\angle C = 2x + 15^{\circ}$. Find the value of $x$.

In a cyclic quadrilateral $ABCD$,$\angle A : \angle C = 4 : 5$ and $\angle B : \angle D = 5 : 7$. Find all the angles of $ABCD$.

State whether each of the following statements is true or false:
$(1)$ $A$ circle divides the plane on which it lies into three parts.
$(2)$ $A$ point,whose distance from the centre of a circle is greater than its radius,lies in the interior of the circle.

In the given figure,if $\angle BCD = 40^{\circ}$ and $\angle BAE = 65^{\circ},$ then find the values of $a, b,$ $c$ and $d$.

In a circle with radius $17 \, cm$,the length of a chord is $30 \, cm$. Find the distance of the chord from the centre. (in $, cm$)

In a circle with centre $P$,$AB$ is a diameter and point $C$ is a point other than $A$ and $B$. If the radius of the circle is $34 \, cm$ and $AC = 32 \, cm$,then $BC = $ .......... $cm$.