1. **State the problem:** We need to find the measure of angle $\angle C$ in triangle $ABC$ where $\angle A = 18^\circ$ and $\angle B = 110^\circ$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles in any triangle is always $180^\circ$. 3. **Write the equation:** $$\angle A + \angle B + \angle C = 180^\circ$$ 4. **Substitute the known values:** $$18^\circ + 110^\circ + \angle C = 180^\circ$$ 5. **Simplify the left side:** $$128^\circ + \angle C = 180^\circ$$ 6. **Isolate $\angle C$:** $$\angle C = 180^\circ - 128^\circ$$ 7. **Calculate the result:** $$\angle C = 52^\circ$$ **Final answer:** $\boxed{52^\circ}$