1. **Problem statement:** Given two parallel lines $\overline{NO} \parallel \overline{PQ}$ and points as described, find the measure of angle $m \angle QSR$. 2. **Identify given angles and relationships:** - $m \angle NRO = (5x + 86)^\circ$ - $m \angle PSQ = (10x + 46)^\circ$ - Lines $NO$ and $PQ$ are parallel, so corresponding or alternate interior angles are equal. 3. **Use parallel lines angle properties:** Since $\overline{NO} \parallel \overline{PQ}$, angles $\angle NRO$ and $\angle PSQ$ are corresponding angles, so: $$ 5x + 86 = 10x + 46 $$ 4. **Solve for $x$:** $$ 5x + 86 = 10x + 46 \\ 86 - 46 = 10x - 5x \\ 40 = 5x \\ x = \frac{40}{5} = 8 $$ 5. **Find $m \angle PSQ$:** $$ m \angle PSQ = 10x + 46 = 10(8) + 46 = 80 + 46 = 126^\circ $$ 6. **Find $m \angle QSR$:** Since $\angle QSR$ and $\angle PSQ$ are supplementary (they form a straight line at point S), $$ m \angle QSR = 180^\circ - m \angle PSQ = 180^\circ - 126^\circ = 54^\circ $$ **Final answer:** $$ m \angle QSR = 54^\circ $$