1. **State the problem:** We have a parallelogram PQRS with $\angle Q = 128^\circ$. We need to find the measure of $\angle S$. 2. **Recall properties of parallelograms:** Opposite angles in a parallelogram are equal, and adjacent angles are supplementary (sum to $180^\circ$). 3. **Apply the supplementary angle rule:** Since $\angle Q$ and $\angle S$ are adjacent angles, we have: $$\angle Q + \angle S = 180^\circ$$ 4. **Substitute the known value:** $$128^\circ + \angle S = 180^\circ$$ 5. **Solve for $\angle S$:** $$\angle S = 180^\circ - 128^\circ = 52^\circ$$ 6. **Conclusion:** The measure of $\angle S$ is $52^\circ$.