
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 Constructing a parallel through a point (rhombus method).
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 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.
Constructions pages on this site
Lines
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
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