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).

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.

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

- 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)

- 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

- 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 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)

- 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

- 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

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