1. **Problem statement:** Given two parallel lines M and N cut by a transversal, find the measure of angle $\angle a$ on line M, knowing the corresponding angle on line N is 55°. 2. **Relevant rule:** When two parallel lines are cut by a transversal, corresponding angles are equal. 3. **Apply the rule:** Since lines M and N are parallel and $\angle a$ corresponds to the 55° angle on line N, we have: $$\angle a = 55^\circ$$ 4. **Conclusion:** The measure of $\angle a$ is 55° because corresponding angles formed by a transversal with parallel lines are congruent.