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

This page shows how to construct a line parallel to a given line through a given point with compass and straightedge or ruler. This construction works by creating a rhombus. Since we know that the opposite sides of a rhombus are parallel, then we have created the desired parallel line. This construction is easier than the traditional angle copy method since it is done with just a single compass setting.

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 creating a rhombus. Since we know that the opposite sides of a rhombus are parallel, then we have created the desired parallel lines.

The diagram below is the final drawing above with the green lines added.

  Argument Reason
1 Line segments RJ, JE, ES, RS are congruent All drawn with the same compass width.
2 RJES is a rhombus A rhombus is a quadrilateral with 4 congruent sides.
7 Lines RS and JE are parallel Opposite sides of a rhombus are always parallel. See Definition of a Rhombus

  - 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.

Acknowledgements

Thanks to Eric Reppun, Sacred Hearts Academy, Honolulu, Hawaii for contributing this construction.

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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