Draw a parallel line

The idea

To draw a line through a point B parallel to a line m with ruler and compass, draw a transversal l through B meeting m, then copy the angle between m and l at B to make equal corresponding angles.

Parallel from corresponding angles

You can draw parallel lines with a ruler and a set square, using equal corresponding angles. Now you have only a ruler and a compass. The line m and a point B are given.

How could the compass help to draw a line through B parallel to m?

• Copy an angle so that corresponding angles are equal
• Draw a circle round B
• Measure the distance from B to m

Draw the parallel line

First we need a transversal: a line l that crosses m. We choose the line through B and A. Then we copy the angle between m and l at B.

Equal corresponding angles

At A the transversal l makes an angle with m. We copied that angle at B, on the same side of l. The two angles are corresponding angles, and they are equal. So the new line n is parallel to m.

To draw a line through B parallel to a line m, draw a transversal l through B meeting m, then copy the angle between m and l at B. The equal corresponding angles make the lines parallel.

• Step | What it does
• Transversal l through B | meets m at A
• Copy the angle at A to B | equal corresponding angles
• Line through B and G | parallel to m

Notes

To draw a line through B parallel to a line m, draw a transversal l through B meeting m, then copy the angle between m and l at B to make equal corresponding angles.

Check yourself

Which pair of angles do we make equal to get the parallel line?

The angle between m and the transversal at A is 65°. The copy at B is the corresponding angle with the new line n. How many degrees is the angle on the other side of l at B, next to it on the straight line?

Answer: 115

The copied angle is 65°. The angle next to it on the straight line is 180° − 65° = 115°.

Leela copies the angle at B but makes it 70°, while the angle at A is 65°. Are m and her new line parallel?

The compass is first opened to some radius and used for the arc at A. Why is the same radius used for the arc at B?

• Vertically opposite angles. Those are equal for any two crossing lines. They cannot show parallel lines.
• Corresponding angles — correct. Yes! The angle at A and the angle in the same position at B.
• Angles on a straight line. They add up to 180°, whatever the lines are.
• Yes. Any line through B is parallel to m.. Only one line through B is parallel to m. Her line leans a little.
• No. The corresponding angles are not equal. — correct. Yes! Parallel lines have equal corresponding angles.
• Yes, because 70° is close to 65°.. Close is not equal. The angles must be exactly equal.
• So that the new line is as long as m.. The lengths of the lines do not matter. It is the angle that is copied.
• So that B and A are the same distance from m.. The radius does not move B or A. It lets us carry the angle across.
• So that the distance CD can be carried across: the two arcs must match. — correct. Yes! The chord CD measures the angle on an arc of this radius, so the copy needs the same radius.