1. **Stating the problem:** We want to understand why $\angle EHF = \angle BED$ due to alternate interior angles, and why $\angle EHF \neq \angle BEH$. 2. **Key concept:** Alternate interior angles are formed when a transversal crosses two parallel lines. These angles lie between the two lines but on opposite sides of the transversal, and they are equal. 3. **Explanation:** If lines $\overline{BE}$ and $\overline{HF}$ are parallel and $\overline{EH}$ is the transversal, then $\angle EHF$ and $\angle BED$ are alternate interior angles, so they are equal. 4. **Why $\angle EHF \neq \angle BEH$:** These two angles are not alternate interior angles; they do not lie on opposite sides of the transversal between the parallel lines. Instead, $\angle BEH$ is adjacent to $\angle EHF$ but not equal by the alternate interior angle theorem. 5. **Summary:** The equality $\angle EHF = \angle BED$ holds because they are alternate interior angles formed by a transversal crossing parallel lines, while $\angle EHF$ and $\angle BEH$ are not alternate interior angles and thus not necessarily equal.