1. **Stating the problem:** We are given two parallel lines $\overrightarrow{AB}$ and $\overrightarrow{CD}$ with transversal lines intersecting them, forming angles. We know $\angle 2 = 58^\circ$ on the top line and $\angle = 61^\circ$ on the bottom line. We need to find $\angle 1$. 2. **Relevant rules and formulas:** When two parallel lines are cut by a transversal, alternate interior angles are equal, and corresponding angles are equal. Also, angles on a straight line sum to $180^\circ$. 3. **Analyze the angles:** $\angle 2 = 58^\circ$ is given on the top line. Since $\overrightarrow{AB} \parallel \overrightarrow{CD}$, the angle corresponding to $\angle 2$ on the bottom line is also $58^\circ$. 4. **Use the given $61^\circ$ angle on the bottom line:** The two angles on the bottom line formed by the transversal are $61^\circ$ and the corresponding $58^\circ$ angle. These two angles are adjacent and form a straight line, so their sum is $180^\circ$. 5. **Calculate the sum:** $$61^\circ + 58^\circ = 119^\circ$$ 6. **Find the missing angle on the top line, $\angle 1$:** Since $\angle 1$ and $\angle 2$ are on a straight line, their sum is $180^\circ$. 7. **Calculate $\angle 1$:** $$\angle 1 + 58^\circ = 180^\circ$$ $$\angle 1 = 180^\circ - 58^\circ = 122^\circ$$ **Final answer:** $\angle 1 = 122^\circ$