Constructing a parallel through a point with compass and straightedge

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Constructing a parallel line through a point

Click here for a printable parallel line construction worksheet

This demonstration shows how to construct a line parallel to a given line through a point, using only a compass and straight edge. See "Introduction to Euclidean Constructions" We start with a line

and a point R. The result is another line

parallel to the first.

Instructions Click on 'Next' to go through the construction one step at a time, or click on 'Run' to let it run without stopping.

(If there is no image below, see support page.)

Note that the straightedge can be a ruler, but it is not used to actually measure anything, just to draw straight lines. Ignore the markings on it.

Step-by-step Instructions

Step 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.
Step 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.
Step 3	Without adjusting the compass width, move the compass to R and draw a similar arc to the one in step 2.
Step 4	Set compass width to the distance between where the lower arc crosses the two lines
Step 5	Move the compass to where the upper arc crosses the transverse line and draw an arc across the upper arc, forming point S.
Step 6	Draw a straight line through points R and S.
Step 6	Done. The line RS is parallel to the line PQ

Explanation

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.

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.

Other constructions

Lines

Angles

Triangles

Triangle Centers

Circles, Arcs and Ellipses

Non-Euclidean constructions