How do you prove theorems behind two parallel lines cut by a transversal?

The way to do it can be explained like this:
Say AB and CD are the two parallel lines cut by a transversal at E and F respectively.
 Then the pairs of alternate interior angles are:
Angle(AEF) and Angle(DFE)
Angle(CFE) and Angle(BEF)
Now lets prove if this is true:
Angle(CFE) +Angle(DFE) = 180
(linear pair)
Also
Angle(CFE) +Angle(AEF) = 180
(Corresponding angles)
Equate the above results:
Angle(CFE) +Angle(DFE) = Angle(CFE) +Angle(AEF)
Angle(DFE) = Angle(AEF)
Happens the same with
Angle(CFE) = Angle(BEF)
Hope this is very useful for you

I'm not sure exactly but my best guess would be 1/4