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.

(TRUE) The given statement is true.
According to the theorem,if a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the segment,then the four points lie on a circle (i.e.,they are concyclic).
Here,the line segment $BC$ subtends equal angles $\angle BAC = 45^{\circ}$ and $\angle BDC = 45^{\circ}$ at points $A$ and $D$ respectively.
Since these points lie on the same side of $BC$,the points $A, B, C$ and $D$ must be concyclic.