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:
- Constructing a parallel through a point (angle copy method).
- Constructing a parallel through a point (rhombus method).
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 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
- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Difference of two line segments
- Perpendicular bisector of a line segment
- Perpendicular from a line at a point
- Perpendicular from a line through a point
- Perpendicular from endpoint of a ray
- Divide a segment into n equal parts
- Parallel line through a point (angle copy)
- Parallel line through a point (rhombus)
- Parallel line through a point (translation)
Angles
- Bisecting an angle
- Copy an angle
- Construct a 30° angle
- Construct a 45° angle
- Construct a 60° angle
- Construct a 90° angle (right angle)
- Sum of n angles
- Difference of two angles
- Supplementary angle
- Complementary angle
- Constructing 75° 105° 120° 135° 150° angles and more
Triangles
- Copy a triangle
- Isosceles triangle, given base and side
- Isosceles triangle, given base and altitude
- Isosceles triangle, given leg and apex angle
- Equilateral triangle
- 30-60-90 triangle, given the hypotenuse
- Triangle, given 3 sides (sss)
- Triangle, given one side and adjacent angles (asa)
- Triangle, given two angles and non-included side (aas)
- Triangle, given two sides and included angle (sas)
- Triangle medians
- Triangle midsegment
- Triangle altitude
- Triangle altitude (outside case)
Right triangles
- Right Triangle, given one leg and hypotenuse (HL)
- Right Triangle, given both legs (LL)
- Right Triangle, given hypotenuse and one angle (HA)
- Right Triangle, given one leg and one angle (LA)
Triangle Centers
Circles, Arcs and Ellipses
- Finding the center of a circle
- Circle given 3 points
- Tangent at a point on the circle
- Tangents through an external point
- Tangents to two circles (external)
- Tangents to two circles (internal)
- Incircle of a triangle
- Focus points of a given ellipse
- Circumcircle of a triangle
Polygons
- Square given one side
- Square inscribed in a circle
- Hexagon given one side
- Hexagon inscribed in a given circle
- Pentagon inscribed in a given circle
Non-Euclidean constructions
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