Drawing a parallel line

Copying an angle has an important application: it is what lets us construct parallel lines with a ruler and compass.

Recall how you drew parallels with a ruler and set square in Part 1. You slid the set square along a line and made two lines perpendicular to it — that is, you constructed equal CORRESPONDING angles. Equal corresponding angles are what make two lines parallel.

So the plan is the same, with different tools. Suppose there is a line m, and we want a line parallel to it.

First, construct a line l that intersects m, meeting it at a point A. Line l will serve as a TRANSVERSAL — to m, and to the parallel line we are about to build.

Choose a point B on l, through which the parallel line is to pass. For the new line to be parallel to m, it must make the SAME corresponding angle with l at B as m makes at A.

And we know how to do that now: copy the angle. Draw arcs of equal radius from A and from B; transfer the chord length from the arc at A to the arc at B; and draw the line n through B and the transferred point.

Since the corresponding angles at A and B are now equal, m is parallel to n. Notice how the chapter has built up: congruence justified the perpendicular bisector, which gave the 90° angle; congruence justified the angle copy; and the angle copy gives parallels. Each construction rests on the one before.