The problem asks to prove that $\angle ABC = \angle DEF$, given that $BA \parallel ED$ and $BC \parallel EF$.

Step-by-step solution:

1. Identify Parallel Lines:

   • $BA \parallel ED$
   • $BC \parallel EF$

2. Transversal and Corresponding Angles: Since $BA \parallel ED$ and $BC \parallel EF$, line $BE$ acts as a transversal in both pairs of parallel lines.

   • Using the parallel line property, $\angle ABC$ corresponds to $\angle EFD$, because when two parallel lines are cut by a transversal, corresponding angles are equal.

3. Final Step: $\angle ABC = \angle DEF$ because of the corresponding angle theorem.

Thus, we have shown that $\angle ABC = \angle DEF$.

Would you like more details, or do you have any further questions?

Here are five related questions:

1. What is the corresponding angle theorem, and how does it work in geometry?
2. How do parallel lines affect angle relationships when a transversal crosses them?
3. Can you explain the difference between corresponding and alternate angles?
4. What other angle theorems apply to parallel lines, aside from corresponding angles?
5. How would the solution change if the lines were not parallel?

Tip: Always identify all parallel lines and transversals in geometry problems to use angle theorems efficiently!