Proof question-asymptotic triangle

Hey guys, there is a proof on a university website on the bottom of the page here:

Strange New Triangles

Now, I'm wondering at the end of the first part of the proof where it shows the three angles are congruent...why this means that $C\phi$ and $D\phi$ are no longer limiting parallels? forgive me for simply providing the link...the proof is very latex heavy.

Thanks guys!