Proofs are an exercise in thinking logically about how you know what you know. They generally assume the existence of some list of theorems and postulates that are taken to be true and can be cited in your proof as reasons for the statements you make. I've found such lists are not often written in one place, but tend to be scattered through geometry texts, and may vary from one proof to the next.
28) The thrust of this proof is to show that angles 2 and 3 are congruent, then cite the Converse of the Corresponding Angles Postulate, which says that when corresponding angles made by a transversal are equal, the lines are parallel. The proof gets to 2 and 3 being congruent by showing they are both supplementary to angle 1. You know this from the clues offered by the reasons that are given.
reason 1: Given
statement 3: ∠1 is supplementary to ∠3
statement 4: ∠2 is congruent to ∠3 . . . . (you have just shown they are both supplementary to ∠1)
reason 5: Converse of the Corresponding Angles Postulate
#?) The second proof is not so different from the first. Here, you're using the transitive property to show that when things are equal to other things, they are equal to each other. Then the same Converse of ... Postulate is invoked to show the lines are parallel.
1: AB ║ CD, ∠2 ≅ ∠1, ∠3 ≅ ∠4 . . . Given
2: ∠1 ≅ ∠3 . . . Corresponding Angles Postulate
3: ∠1 ≅ ∠4 . . . Transitive property (both are congruent to ∠3)
4: ∠2 ≅ ∠4 . . . Transitive property (both are congruent to ∠1)
5: BC ║ DE . . . Converse of Corresponding Angles Postulate
