1. **Problem statement:** Find the measure of angle $x$ given two parallel lines $a \parallel b$ cut by a transversal, with angles $144^\circ$ and $36^\circ$ marked as shown. 2. **Relevant rules:** When two parallel lines are cut by a transversal, alternate interior angles are equal, corresponding angles are equal, and consecutive interior angles are supplementary (sum to $180^\circ$). 3. **Analyze given angles:** The angle $144^\circ$ and $36^\circ$ are supplementary because they form a linear pair on the same side of the transversal. 4. **Find angle $x$:** Since $a \parallel b$, angle $x$ corresponds to the angle adjacent to $36^\circ$ on line $b$. Because $36^\circ$ is an interior angle on line $a$, angle $x$ is its alternate interior angle and thus equal to $36^\circ$. 5. **Final answer:** $$x = 36^\circ$$