Constructing Parallel Lines

The Parallel Postulate states that exactly one line through a point P exists that is parallel to a line l. The line through P can be found using a compass and a straightedge.

The parallel line can be constructed in several steps.

Draw a Line Through P and Across l at an Angle

Begin by drawing a line that intersects l, at any arbitrary location, and goes through the point. The point where the new line intersects l will be named Q.

Transfer a Copy of the Angle to P

A copy of the angle formed by l and QP will be constructed in order to transfer the copy to point P. To copy the angle, begin by fixing the needle of the compass at Q. With the compass set to an arbitrary length, draw an arc on QP between Q and P. Then, with the same setting, draw an arc from P across QP.

Adjust the radius of the compass so that both compass ends are on the first drawn arc's intersection points with l and QP. Using that setting, place the needle of the compass where the second arc intersects QP. Draw an arc intersecting the second arc. The point where these arcs intersect can be labeled R.

Draw a Line Through P and R

Draw the line that connects points P and R.

This line is parallel to l.

Recommended exercises