1. **State the problem:** We need to solve for $x$ given that the angle at point N is $(5x - 3)^\circ$ and the angle at point O is $62^\circ$. 2. **Understand the relationship:** Since the two lines intersect, the angles around the intersection points are related. Typically, vertically opposite angles are equal. 3. **Set up the equation:** Assuming $(5x - 3)^\circ$ and $62^\circ$ are vertically opposite angles, we have: $$5x - 3 = 62$$ 4. **Solve for $x$:** $$5x - 3 = 62$$ Add 3 to both sides: $$5x - \cancel{3} + 3 = 62 + 3$$ $$5x = 65$$ Divide both sides by 5: $$\frac{5x}{\cancel{5}} = \frac{65}{5}$$ $$x = 13$$ 5. **Final answer:** $$\boxed{13}$$