1. **State the problem:** We are given two parallel lines cut by a transversal, creating angles $ (9x + 31)^\circ $ at point Q and $ (5x + 47)^\circ $ at point R. We need to find the measure of $ \angle QRO $. 2. **Identify the relationship:** Since the lines are parallel and $ Q $ and $ R $ lie on the transversal, the angles $ \angle MQN = 9x + 31 $ and $ \angle ORP = 5x + 47 $ are corresponding angles or alternate interior angles, which are equal. 3. **Set up the equation:** $$ 9x + 31 = 5x + 47 $$ 4. **Solve for $ x $:** $$ 9x + 31 = 5x + 47 \\ 9x - 5x = 47 - 31 \\ \cancel{9}x - \cancel{5}x = 16 \\ 4x = 16 \\ \frac{\cancel{4}x}{\cancel{4}} = \frac{16}{4} \\ x = 4 $$ 5. **Find $ \angle QRO $:** Substitute $ x = 4 $ into $ 5x + 47 $: $$ 5(4) + 47 = 20 + 47 = 67 $$ 6. **Answer:** $ m\angle QRO = 67^\circ $. This means the measure of angle QRO is 67 degrees.