1. **State the problem:** We need to find the measure of angle $\angle ORQ$ given two expressions for angles formed by intersecting lines: $(9x - 4)^\circ$ and $(3x + 40)^\circ$. 2. **Understand the geometry:** The lines intersect at points $Q$ and $R$, forming vertical angles. Vertical angles are equal. 3. **Set up the equation:** Since the angles are vertical angles, we have: $$9x - 4 = 3x + 40$$ 4. **Solve for $x$:** $$9x - 3x = 40 + 4$$ $$6x = 44$$ $$x = \frac{44}{6} = \frac{22}{3} \approx 7.33$$ 5. **Find $\angle ORQ$:** Substitute $x$ back into one of the angle expressions, for example, $3x + 40$: $$3 \times \frac{22}{3} + 40 = 22 + 40 = 62^\circ$$ 6. **Conclusion:** The measure of $\angle ORQ$ is $62^\circ$. This uses the property that vertical angles are equal and basic algebra to solve for $x$ and then find the angle measure.