1. **Problem Statement:** Given two parallel lines $\overrightarrow{IK}$ and $\overrightarrow{LN}$, and a transversal intersecting them at points $J$ and $M$ respectively, with $m \angle IJH = 128^\circ$, find $m \angle KJM$. 2. **Relevant Concept:** When a transversal crosses two parallel lines, alternate interior angles are equal, and consecutive interior angles are supplementary (sum to $180^\circ$). 3. **Identify Angles:** $\angle IJH$ and $\angle KJM$ are on the same side of the transversal and between the two parallel lines, so they are consecutive interior angles. 4. **Apply Supplementary Angle Rule:** Since $\angle IJH$ and $\angle KJM$ are consecutive interior angles, $$m \angle IJH + m \angle KJM = 180^\circ$$ 5. **Calculate $m \angle KJM$:** $$m \angle KJM = 180^\circ - m \angle IJH = 180^\circ - 128^\circ = 52^\circ$$ 6. **Answer:** $$m \angle KJM = 52^\circ$$