In the figure,if $\angle ABC = 20^{\circ},$ then $\angle AOC$ is equal to (in $^{\circ}$)

$AD$ is a diameter of a circle and $AB$ is a chord. If $AD = 34 \, cm$ and $AB = 30 \, cm$,the distance of $AB$ from the centre of the circle is (in $cm$):

If non-parallel sides of a trapezium are equal,prove that it is cyclic.

Prove that equal chords of congruent circles subtend equal angles at their centres.

Two circles of radii $8 \ cm$ with centers $P$ and $Q$ are given. The chord $AB$ of the circle with center $P$ and the chord $CD$ of the circle with center $Q$ are equal. If $\angle PAB = 40^{\circ}$,then find $\angle CQD$. (in $^{\circ}$)

State whether each of the following statements is true or false:
$(1)$ There are three circles passing through three given non-collinear points.
$(2)$ Infinitely many circles pass through two given points.