How to construct (draw) a parallel through a point with compass and straightedge or ruler

Constructing a parallel through a point

Geometry construction using a compass and straightedge

Step-by-step Instructions

After doing this	Your work should look like this
Start with a line segment PQ and a point R off the line.
1. Draw a transverse line through R and across the line PQ at an angle, forming the point J where it intersects the line PQ. The exact angle is not important.
2. With the compass width set to about half the distance between R and J, place the point on J, and draw an arc across both lines.
3. Without adjusting the compass width, move the compass to R and draw a similar arc to the one in step 2.
4. Set compass width to the distance where the lower arc crosses the two lines.
5. Move the compass to where the upper arc crosses the transverse line and draw an arc across the upper arc, forming point S.
6. Draw a straight line through points R and S.
Done. The line RS is parallel to the line PQ

Proof

This construction works by using the fact that a transverse line drawn across two parallel lines creates pairs of equal corresponding angles. It uses this in reverse - by creating two equal corresponding angles, it can create the parallel lines.

The image below is the final drawing above with the red items added.

	Argument	Reason
1	Line segments AR,BJ are congruent	Both drawn with the same compass width.
2	Line segments RS,JC are congruent	Both drawn with the same compass width.
3	Line segments AS,BC are congruent	Both drawn with the same compass width.
4	Triangles ∆ARS and ∆BJC are congruent	Three sides congruent (SSS).
5	Angles ARS, BJC are congruent.	CPCTC. Corresponding parts of congruent triangles are congruent
6	The line AJ is a transversal	It is a straight line drawn with a straightedge and cuts across the lines RS and PQ.
7	Lines RS and PQ are parallel	Angles ARS, BJC are corresponding angles that are equal in measure only if the lines RS and PQ are parallel

- Q.E.D

Try it yourself

Click here for a printable parallel line construction worksheet containing two problems to try. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.

Proof

Constructions pages on this site

Lines

Angles

Triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions