constructing a parallel through a point
Given a line and a point not on it, you want the line through that point that runs forever alongside the first without ever meeting it. Set squares slide one against the other to do this in a drafting class; with compass and straightedge you reach the same parallel through a small chain of the basic moves.
One clean method copies an angle. Draw any line through the point P that also crosses the given line; this transversal makes some angle with the given line. Now copy that exact angle at P, on the same side, opening the same way. By the parallel-lines criterion, equal corresponding angles force the two lines to be parallel, so the new ray through P never meets the original. A second method drops a perpendicular from P to the line, then erects a perpendicular to THAT perpendicular at P — two right turns leave you parallel to where you started.
This construction is the constructive face of Euclid's parallel postulate and of Playfair's axiom (through a point not on a line there is exactly one parallel). In ordinary Euclidean geometry the parallel exists and is unique; in hyperbolic geometry that uniqueness fails, which is why the parallel postulate is independent rather than provable. Constructing parallels underlies dividing a segment into equal parts, building parallelograms, and setting up coordinate grids.
Through P, draw a transversal crossing the given line and copy the angle it makes there to P on the matching side. Equal corresponding angles make the new line through P parallel to the original.
Copying a transversal's angle forces the corresponding angles equal — and equal corresponding angles mean parallel.
Uniqueness of the parallel is the content of Playfair's axiom, which is equivalent to Euclid's parallel postulate; it holds in Euclidean geometry but fails in hyperbolic geometry, so the parallel postulate is independent, not a theorem.