1. The problem asks: Which angle is complementary to $\angle 5$? 2. Complementary angles are two angles whose measures add up to $90^\circ$. 3. In the diagram of two intersecting lines, angles formed around the intersection are either vertical angles or adjacent angles. 4. Since $\angle 5$ is at the intersection, we look for an angle adjacent to $\angle 5$ that, together with $\angle 5$, sums to $90^\circ$. 5. Typically, angles formed by intersecting lines are either supplementary ($180^\circ$) or vertical (equal in measure), so complementary angles are usually adjacent angles that form a right angle. 6. From the given angles $\angle 4$, $\angle 6$, $\angle 2$, and $\angle 3$, the angle adjacent to $\angle 5$ that forms a right angle is $\angle 4$. 7. Therefore, $\angle 4$ is complementary to $\angle 5$. Final answer: $\boxed{\angle 4}$