$ABCD$ is a cyclic quadrilateral in which $\angle BAC = 45^{\circ}$ and $\angle ADB = 55^{\circ},$ then $\angle ABC = \dots$ (in $^{\circ}$)

Prove that if chords of congruent circles subtend equal angles at their centres,then the chords are equal.

In a cyclic quadrilateral $PQRS$,$\angle QPR = 40^{\circ}$ and $\angle PSQ = 70^{\circ}$,then find $\angle PQR$. (in $^{\circ}$)

$AB$ and $CD$ are two parallel chords of a circle with centre $P$. $l$ is the perpendicular bisector of chord $AB$. Prove that $l$ bisects chord $CD$.

Two circles with centres $O$ and $O'$ intersect at two points $A$ and $B$. $A$ line $PQ$ is drawn parallel to $OO'$ through $A$ (or $B$) intersecting the circles at $P$ and $Q$. Prove that $PQ = 2 OO'$.

Two chords $AB$ and $AC$ of a circle subtend angles equal to $90^{\circ}$ and $150^{\circ}$,respectively,at the centre. Find $\angle BAC$,if $AB$ and $AC$ lie on the opposite sides of the centre.