Constructing parallel lines with a compass and straightedge uses the converse of the parallel lines theorem. Creating congruent corresponding angles (or congruent AIA or AEA) guarantees parallel lines. The first step in constructing parallel lines is to draw a transversal through the given point to intersect the given line. Last, duplicate an angle created by the transversal and the given line.
When you're asked to construct a line through a point parallel to a given line, you are going to use three different methods. You could either choose corresponding angles, alternate interior angles or alternate exterior angles. So you are going to duplicate one of these three pairs of angles. But why does that work? Well, to do that let's take a little sketch.
If I had two lines and if I intersect these two lines by a transversal and I tell you that one pair of corresponding angles are congruent. Then the converse of the parallel lines theorem says that these two lines must be parallel. So this will also work if you have alternate exterior angles. So if you have angles that are on opposite sides of the transversal, or if you have alternate interior angles that are congruent. So you've three different ways of constructing parallel lines through a point.