Then ∠D=∠AFE, because ∠D+∠C= 180, ∠AFE+∠C= 180.

And because AFE+BFE = 180, so

∠BFE=∠C, then it is necessary to prove △ FBE △ CBE.

So BC=FB and AF+FB=AB, so AD+BC=AB.

This should be the easiest way!