1. **State the problem:** We need to find the size of angle $BCD$ in the given figure. 2. **Analyze the given information:** The angle at vertex $B$ is given as $128^\circ$. 3. **Use the property of angles on a straight line:** Since points $A$, $B$, $C$, and $D$ lie on a straight line, the sum of angles around point $B$ on the line is $180^\circ$. 4. **Calculate angle $ABC$:** $$\text{Angle } ABC = 180^\circ - 128^\circ = 52^\circ$$ 5. **Use triangle angle sum property:** In triangle $BCD$, the sum of interior angles is $180^\circ$. 6. **Assuming angle $CBD$ is the same as angle $ABC$ (since $B$ is common and line is straight), then:** $$\text{Angle } BCD = 180^\circ - 90^\circ - 52^\circ = 38^\circ$$ 7. **Final answer:** $$\boxed{38^\circ}$$ This means the size of angle $BCD$ is $38$ degrees.