Use the converse angle theorems to prove two lines are parallel, and apply the Parallel Postulate.
Last week assumed two lines were parallel and found angle measures. This week runs the logic backward: showing that one pair of corresponding angles is congruent (or alternate interior, or alternate exterior), or that a pair of same-side interior angles is supplementary, is enough to PROVE the two lines are parallel. These are the converses of last week's theorems, and a converse is never automatically true just because the original theorem is -- each one had to be proved separately. In practice, this is how you show two lines are parallel without measuring the whole figure: find or create just one matching angle pair.
Two more facts round out the toolbox. The Parallel Postulate states that through a point not on a given line, there is exactly one line parallel to that line -- the assumption underneath this whole unit, and the reason parallel lines never converge no matter how far they extend. A short chain of reasoning gives two more useful theorems: if two lines are each parallel to a third line, they are parallel to each other (Transitive Property of Parallel Lines), and if a transversal is perpendicular to one of two parallel lines, it is automatically perpendicular to the other.
Mastery looks like: They correctly apply the converse theorems to prove two lines parallel from a single angle relationship, and apply the transitive and perpendicular-transversal facts about parallel lines.
Common stumbles: Treating a same-side interior pair as if it must be congruent rather than supplementary; assuming any two congruent angles anywhere prove lines parallel without checking they are a valid corresponding, alternate interior, or alternate exterior pair.