1. **Problem statement:** Find the measure of angle $\angle BDC$ in triangle $BCD$ where $\angle CBD = 60^\circ$ and $\angle C = 90^\circ$. 2. **Recall the angle sum property of a triangle:** The sum of interior angles in any triangle is $180^\circ$. 3. **Apply the property to triangle $BCD$:** $$\angle B + \angle C + \angle D = 180^\circ$$ Here, $\angle B = \angle CBD = 60^\circ$, $\angle C = 90^\circ$, and $\angle D = \angle BDC$ (unknown). 4. **Calculate $\angle BDC$:** $$\angle BDC = 180^\circ - \angle CBD - \angle C = 180^\circ - 60^\circ - 90^\circ = 30^\circ$$ 5. **Answer:** The measure of $\angle BDC$ is $30^\circ$.