
Constructing a parallel through a point (rhombus method)
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 stepbystep instructions
The above animation is available as a
printable stepbystep 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.
Other constructions pages on this site
Lines
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
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