1. **State the problem:** Given two parallel lines $p$ and $q$ cut by a transversal, we are to understand the relationships between angles $\angle 1$, $\angle 2$, and $\angle 3$. 2. **Identify angle relationships:** - $\angle 1$ and $\angle 2$ are vertical angles. - $\angle 2$ and $\angle 3$ are corresponding angles. - $\angle 1$ and $\angle 3$ are alternate exterior angles. 3. **Recall angle rules:** - Vertical angles are congruent: $m\angle 1 = m\angle 2$. - Corresponding angles formed by parallel lines and a transversal are congruent: $m\angle 2 = m\angle 3$. - Alternate exterior angles formed by parallel lines and a transversal are congruent: $m\angle 1 = m\angle 3$. 4. **Given:** $m\angle 2 = 55^\circ$. 5. **Apply vertical angle rule:** $$m\angle 1 = m\angle 2 = 55^\circ$$ 6. **Apply corresponding angle rule:** $$m\angle 3 = m\angle 2 = 55^\circ$$ 7. **Conclude:** Since $m\angle 1 = m\angle 3 = 55^\circ$, $\angle 1$ and $\angle 3$ are congruent alternate exterior angles, confirming the rule that when parallel lines are cut by a transversal, alternate exterior angles are congruent. **Final answer:** $$m\angle 1 = m\angle 2 = m\angle 3 = 55^\circ$$