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