1. **State the problem:** We need to find the measure of angle $\angle MPQ$ given two expressions for angles formed by a transversal intersecting two parallel lines. 2. **Identify the angles:** The angles given are: - At point $P$: $(9x - 35)^\circ$ - At point $Q$: $(4x + 30)^\circ$ 3. **Use the property of parallel lines and a transversal:** When a transversal crosses two parallel lines, alternate interior angles are equal. Here, $\angle MPQ$ corresponds to the angle at $Q$, so these two angles are equal: $$9x - 35 = 4x + 30$$ 4. **Solve for $x$:** $$9x - 35 = 4x + 30$$ $$9x - \cancel{35} - 4x = 4x + 30 - \cancel{35}$$ $$5x = 65$$ $$x = \frac{65}{5}$$ $$x = 13$$ 5. **Find $\angle MPQ$ by substituting $x=13$ into one of the angle expressions:** $$\angle MPQ = 4x + 30 = 4(13) + 30 = 52 + 30 = 82^\circ$$ **Final answer:** $$m\angle MPQ = 82^\circ$$