
Constructing a parallel through a point
(translated triangle 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 any triangle between the given point and the
given line, then copying
(translating)
that triangle any distance along the given line. Since we know that a translation
can map the one triangle onto the second
congruent triangle,
then the lines linking the corresponding
points of each triangle are parallel, and we can create the desired parallel line by linking the top
vertices of the two triangles.
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 triangle and then translating (sliding) the triangle along the given line.
All corresponding vertices of a translated polygon are linked by lines that are congruent and parallel.
This can be seen more clearly
in the animation at
Translating a Polygon.
(In that animation, check the "Show Lines" box ).

Argument 
Reason 
1 
Triangle ARB and A'R'B' are congruent 
By construction. A'R'B' was copied from ARB. For method and proof see
Copying a triangle 
2 
AB and A'B' are
collinear. 
All four points lie on PQ 
3 
A'R'B' is a translation of ARB. 
The two triangles are congruent
(from 1) and not
rotated
(from 2) and not
reflected (by construction). 
4 
RR' is parallel to AA' 
Lines linking the corresponding vertices of translated polygons are parallel.
See Properties of translated polygons 
5 
RR' is parallel to PQ 
From (2)  AA' is parallel to PQ because they are
collinear. 
 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 Sue Connolly, Irondequoit High School, New York 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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