1. **State the problem:** Given two parallel lines $m \parallel n$, and the measure of angle $\angle 3 = 58^\circ$, find the measure of angle $\angle 7$. 2. **Recall the properties of parallel lines and angles:** When two parallel lines are cut by a transversal, corresponding angles are equal. Also, alternate interior angles are equal. 3. **Identify the relationship between $\angle 3$ and $\angle 7$:** Since $m \parallel n$, and $\angle 3$ and $\angle 7$ are corresponding angles (or alternate interior angles depending on the figure), they have the same measure. 4. **Write the equation:** $$\angle 7 = \angle 3$$ 5. **Substitute the known value:** $$\angle 7 = 58^\circ$$ 6. **Conclusion:** The measure of $\angle 7$ is $58^\circ$.