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Constructing a parallel through a point (angle copy method)
Geometry construction using a compass and straightedge

This page shows how to construct a line parallel to a given line that passes through a given point with compass and straightedge or ruler. It is called the 'angle copy method' because it 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.

See also:

Printable step-by-step instructions

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

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.
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Other constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions