If arcs $AXB$ and $CYD$ of a circle are congruent,find the ratio of $AB$ and $CD.$

Write True or False and justify your answer in each of the following:
If $A, B, C$ and $D$ are four points such that $\angle BAC = 45^{\circ}$ and $\angle BDC = 45^{\circ}$,then $A, B, C, D$ are concyclic.

In a circle with centre $P$,$AB$ and $CD$ are equal chords. If $\angle APB = 80^{\circ}$,then find $\angle CPD$. (in $^{\circ}$)

If a pair of opposite sides of a cyclic quadrilateral are equal,prove that its diagonals are also equal.

In a circle with centre $P$,$AB$ is a chord and $PA = 4\, cm$. In a circle with centre $Q$,$XY$ is a chord and $QX = 4\, cm$. If $\angle APB = 80^{\circ}$,$\angle XQY = 50^{\circ}$ and $AB = 5\, cm$,then $XY = \dots\dots\dots\, cm$.

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 $^{\circ}$)