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

Write True or False and justify your answer in each of the following: Two chords $AB$ and $CD$ of a circle are each at distances $4 \ cm$ from the centre. Then $AB = CD$.

$ABCD$ 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 the figure,two congruent circles have centers $O$ and $O'$. Arc $AXB$ subtends an angle of $75^{\circ}$ at the center $O$ and arc $A'YB'$ subtends an angle of $25^{\circ}$ at the center $O'$. Then the ratio of the lengths of arcs $AXB$ and $A'YB'$ is:

$ABCD$ is a cyclic quadrilateral such that $AB$ is a diameter of the circle circumscribing it and $\angle ADC = 140^{\circ}$,then $\angle BAC$ is equal to (in $^{\circ}$)

$ABCD$ is a parallelogram. $A$ circle through $A$ and $B$ is drawn such that it intersects $AD$ at $P$ and $BC$ at $Q$. Prove that $P, Q, C,$ and $D$ are concyclic.