Use a paragraph proof to prove the corresponding angles postulate:

given: segment a b is parallel to segment c d.

prove: corresponding angles are congruent

transversal ef cuts through parallel lines segments ab and cd creating four sets of corresponding angles; angles 1, 2, 3, 4 are formed by ef and ab; angles 5, 6, 7, 8 are formed by the ef and cd.

We know that vertical angles are congruent and corresponding angles are congruent if they are formed by parallel lines.
Therefore, 
We can use this information to prove alternate interior angles are also congruent if they are formed by parallel lines 

since the lines are parallel,
then 
< 1 and < 3 are congruent, 
< 2 and < 3 are vertical angles 

We also know that all vertical angles are congruent, so
that means < 1 congruent to < 3 because they are corresponding angles, 
and < 2 congruent to < 3 because they are vertical angles, 
that means < 1 must be congruent to < 3