1. **Problem Statement:** Given two parallel lines $p$ and $q$ cut by a transversal $w$, with $m\angle 1 = 55^\circ$, find the measures of $\angle 2$ and $\angle 3$ and explain the relationships between these angles. 2. **Key Angle Relationships:** - Vertical angles are congruent. - Alternate interior angles are congruent when lines are parallel. - Corresponding angles are congruent when lines are parallel. 3. **Step 1: Vertical Angles** Since $\angle 1$ and $\angle 2$ are vertical angles, they are congruent. $$m\angle 2 = m\angle 1 = 55^\circ$$ 4. **Step 2: Alternate Interior Angles** Since lines $p$ and $q$ are parallel, $\angle 2$ and $\angle 3$ are alternate interior angles and thus congruent. $$m\angle 3 = m\angle 2 = 55^\circ$$ 5. **Step 3: Corresponding Angles** $\angle 1$ and $\angle 3$ are corresponding angles formed by the transversal $w$ with parallel lines $p$ and $q$. By the corresponding angles postulate, they are congruent. $$m\angle 1 = m\angle 3 = 55^\circ$$ 6. **Summary:** - $m\angle 1 = 55^\circ$ - $m\angle 2 = 55^\circ$ (vertical angles) - $m\angle 3 = 55^\circ$ (alternate interior and corresponding angles) This confirms the rule: *When parallel lines are cut by a transversal, corresponding angles are congruent.*