To find the measure of \(\angle DBC\), we can use the relationship between angles and arcs in a circle.
Given that:
- \(BD\) is the diameter of circle \(A\), it means that angle \(\angle BCD\) will be a right angle (90 degrees), because an angle inscribed in a semicircle is always a right angle.
- The measure of arc \(CB\) is 36 degrees.
Using the fact that the measure of an angle inscribed in a circle is half the measure of the arc that it subtends, we have:
\[ \angle DBC = \frac{1}{2} \times \text{measure of arc CB} \]
Since the measure of arc \(CB\) is 36 degrees:
\[ \angle DBC = \frac{1}{2} \times 36 = 18 \text{ degrees} \]
Therefore, the measure of \(\angle DBC\) is \( \boxed{18^\circ} \).