How do I know if this quadrilateral is cyclic?

Question

There are some pairs of angles that are equal to each other, but none of the values are known.

Answer 1

The qudrilaterial is cyclic because the two congruent angles $\angle DCA$ and $\angle DBA$ are inscribed angles from the chord $DA$.

Thus you can use Brahmagupta's formula for the area.

Answer 2

A convex quadrilateral polygon in a plane is cyclic if, and only if, its opposite angles are supplementary.