1. **State the problem:** We have an acute triangle $\triangle XYZ$ with angles $\angle X = 50^\circ$ and $\angle Y = 60^\circ$. The altitudes from vertices $X$ and $Y$ intersect at the orthocenter $H$. We need to find the measure of $\angle XHY$. 2. **Recall properties:** The orthocenter $H$ is the intersection of the altitudes of a triangle. A key property is that the angle formed at the orthocenter by two vertices relates to the angles of the triangle as follows: $$\angle XHY = 180^\circ - \angle Z$$ where $\angle Z$ is the angle at vertex $Z$ of the triangle. 3. **Find $\angle Z$:** Since the sum of angles in a triangle is $180^\circ$, $$\angle Z = 180^\circ - \angle X - \angle Y = 180^\circ - 50^\circ - 60^\circ = 70^\circ$$ 4. **Calculate $\angle XHY$:** Using the property, $$\angle XHY = 180^\circ - 70^\circ = 110^\circ$$ 5. **Conclusion:** The measure of $\angle XHY$ is $110^\circ$.