1. **Problem statement:** Given two parallel lines $p$ and $q$ cut by a transversal, with $m\angle 9 = 129^\circ$, find $m\angle 1$. 2. **Relevant rule:** When a transversal crosses parallel lines, alternate interior angles are equal. 3. Since $\angle 9$ and $\angle 1$ are alternate interior angles, we have: $$m\angle 1 = m\angle 9$$ 4. Substitute the given value: $$m\angle 1 = 129^\circ$$ 5. **Final answer:** $$\boxed{129^\circ}$$