Angles Formed by Parallel Lines
Angle A and angle C are corresponding angles,  angle A and angle D are alternate exterior angles, the angel that is vertical to angle B and the angle that is vertical to angle D are the alterante interior angles.
If you have two parallel lines with a transversal going through both of them, then you can form three types of angles.  Corresponding angles, Alternate inteterior angles and Alternate exterior angles. 
Theorem:  If two parallel lines are cut by a transversal then the alternate interior angles are congruent.  Here is how we can prove this theorem.

Given:  Lines a and b are parallel.
Prove:  angle 1 is congruent to angle 2

To prove this theorem we are going to use an indirect proof.
Assume that angle 1 is not congruent to angle 2.  Then there must be another line c, that intersects the transversal at p to form an angle, angle 3, that is congruent to angle 2, but alternate interior angles lead to parallel lines thus c is parallel to b, then line b is parallel to two lines in the plane.  this contradicts the parallel postulate which states there can be only one parallel line through a point not on the line, therefore angle 1 is congruent to angle 2. 

Theorem: If two parallel lines are cut by a transversal then the alternate exterior angles are congruent.  (The proof for this theorem is similiar to the one above)

Theorem: If two parallel lines are cut by a transversal then the corresponding angles are congruent.  Knowing that the alternate exterior angles are congruent and the alternate interior angles are congruent, you can use algebra to prove this theorem.
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