In a cyclic quadrilateral $ABCD$,$\angle A = 2x - 10^{\circ}$ and $\angle C = 3x - 35^{\circ}$,then $\angle A =$ .......... (in $^{\circ}$)

The longest chord of a circle is a .......... of the circle.

State whether each of the following statements is true or false:
$(1)$ The collection of all the points in a plane,which are at a fixed distance from a fixed point in the plane,is called a circle.
$(2)$ The line segment joining the centre and any point on the circle is called a radius of the circle.

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

$A$ circular park of radius $20\,m$ is situated in a colony. Three boys Ankur,Syed and David are sitting at equal distance on its boundary,each having a toy telephone in his hands to talk with each other. Find the length of the string of each phone.

$A$ quadrilateral $ABCD$ is inscribed in a circle such that $AB$ is a diameter and $\angle ADC = 130^{\circ}$. Find $\angle BAC$.