Constructing a Parallel Through a Point

Parallel lines are defined by an angle condition, not by eyeballed slant. If a transversal crosses two lines and makes equal corresponding angles at both crossings, the lines never meet, no matter how far you extend them. So to build a parallel line through a point P, you first draw any transversal through P that crosses the given line m. That transversal makes some angle with m. Copy that exact angle at P, using the transversal as one ray of the new angle, and the other ray you draw is guaranteed parallel to m.

The copying itself is a compass move, not a protractor guess. Put the compass point on the original vertex and swing one arc that crosses both rays of the angle. Without changing the radius, swing the same arc at the new vertex on the transversal. Then measure the straight-line distance between the two points where the first arc crossed the original rays, and transfer that span to the new arc. Where it lands fixes the second ray, and the angle is reproduced exactly, down to the limits of your drawing.

This sits next to copying segments and bisecting angles in the same toolkit: all three build on the idea that a compass preserves distance exactly, so any length or angle can be carried to a new location without measuring it in degrees or centimeters. Parallel construction also sets up later proofs about transversals, since corresponding or alternate interior angles being equal is the test you will cite to justify why two lines are parallel.