1. **Problem statement:** We have a quadrilateral ABCD with angles $\angle A = 60^\circ$ and $\angle B = 50^\circ$. We need to find the measures of angles $\angle C$ and $\angle D$. 2. **Formula used:** The sum of interior angles in any quadrilateral is always $360^\circ$. This is given by: $$\angle A + \angle B + \angle C + \angle D = 360^\circ$$ 3. **Substitute known values:** $$60^\circ + 50^\circ + \angle C + \angle D = 360^\circ$$ Simplify the known angles: $$110^\circ + \angle C + \angle D = 360^\circ$$ 4. **Express sum of unknown angles:** $$\angle C + \angle D = 360^\circ - 110^\circ = 250^\circ$$ 5. **Additional information needed:** Without more information (such as the quadrilateral being cyclic, or specific side lengths or angle relationships), we cannot determine the individual values of $\angle C$ and $\angle D$ uniquely. **Final answer:** The sum of angles $\angle C$ and $\angle D$ is $250^\circ$, but their individual measures cannot be determined from the given information.