Looking at the proof of Euclid Book 1 Proposition 16, it seems to me that if we were to know for certain that CF, AB were parallel, then the equality of ∠BAE and ∠ECF would follow from them being alternate angles formed when a line crosses two parallel lines.
I'm pretty confident that CF, AB are parallel, but I'm sure sure how to prove the assertion. Also, I wonder if there is any reason to prefer the original proof over this approach if it is valid?