Label something with a variable

In the problem below, it seems we are stuck right away.

However, notice that the base angles of triangle \(BCA\) are congruent. Also, notice that the base angles of triangle \(BCD\) are congruent. Focusing on angle \(DCB,\) you may notice the base angles of triangle \(BCD\) are equal to the angles of triangle \(BCA,\) plus some value. Let's call that unknown value \(y.\) After making this creative step, the rest of the solution falls out. Many puzzles can be solved in this way.