Measure of Angle ACD in Triangle ABC

In triangle ABC, angle A measures $60^{\circ}$. The segment BC is extended to point D, forming an exterior angle at vertex C. The measure of angle DCA is equal to the measure of angle CBA. If the measure of angle CBA is represented as $x$, what is the measure of angle ACD?

Determine the possible measures of angles DCA and ACD based on the relationship of angles in a triangle and external angles. Express your answer in degrees.