I can't quite remember the name of the theorem, but I know it exists. Remember that the picture I attached holds true. That is essentially what is happening with both of your questions, only with parallel lines. Notice that EF is cutting through the parallel lines of FD and AC. Are you following?
∠C≅∠EFD is what I actually drew in there, and ∠CEF≅∠FDC is the same theorem, just with the opposing (smaller) angles. Essentially, you just have to state the theorem and draw in what I did (at least that's what I assume you have to do. I can't imagine you have to prove the theorems too).
Well, if you're just turning in a typed paper, then use the T proof I showed you. However, if you have to enter your answer in a dialog line and press "OK" or something, I'd just say the theorems you used... are you able to take a screen shot of what you're seeing on the site?
The ones you used are the alternate (or Z) angles and the opposite angles
This should be easy now. You can proove $\displaystyle \angle C \equiv \angle EFD$ directly from what BariMutation told you.
And using the fact that CDFE is a rhombus (or using the parallel lines again), you will see that $\displaystyle \angle CEF\equiv\angle FDC.$