1. **Problem Statement:** Given two parallel lines $l \parallel m$, with $\angle 1 = 125^\circ$ and $\angle 7 = 50^\circ$, find the measure of $\angle 5$. 2. **Key Concepts:** When two lines are parallel, alternate interior angles and corresponding angles have special relationships. Also, the sum of angles around a point or in a triangle is important. 3. **Step 1: Analyze $\angle 1$ and $\angle 5$** Since $l \parallel m$, and $\angle 1$ is given as $125^\circ$, $\angle 5$ is an exterior angle adjacent to $\angle 6$ (the top interior vertex angle of the triangle). 4. **Step 2: Use the fact that $\angle 1$ and $\angle 5$ are supplementary** Because $\angle 1$ and $\angle 5$ form a linear pair along line $l$, their measures add up to $180^\circ$: $$\angle 1 + \angle 5 = 180^\circ$$ 5. **Step 3: Substitute the known value of $\angle 1$** $$125^\circ + \angle 5 = 180^\circ$$ 6. **Step 4: Solve for $\angle 5$** $$\angle 5 = 180^\circ - 125^\circ = 55^\circ$$ 7. **Step 5: Verify with $\angle 7$** $\angle 7 = 50^\circ$ is given, which is consistent with the triangle's interior angles and parallel lines but does not affect $\angle 5$ directly. **Final answer:** $$\boxed{55^\circ}$$