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).
- Constructing a parallel through a point (Translated triangles 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 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.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 at a point on a line
- 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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