Parallel Lines

Parallel lines are separate straight lines in the same plane that never intersect. They continue in both directions, even though a drawing can show only part of each line. Their perpendicular separation stays the same.

Look beyond a short drawing

Two short strokes can be apart on the page even when their continuing lines meet farther away. Being apart in one picture is not enough to prove that lines are parallel. Look at their directions and imagine extending them. Arrowheads on a line drawing indicate continuation, not endpoints.

Turning both parallel lines together does not change their relationship. They can be horizontal, vertical or slanted. Two crossing lines cannot be parallel in the same plane. Lines that meet at a right angle are perpendicular; other intersections need not be right angles.

Models around us

Opposite edges of a rectangle lie on parallel lines. The real edges are finite line segments; the mathematical lines containing them continue beyond the corners. The ruled lines on a flat notebook page can model parallel lines too.

A straight section of railway track can suggest parallel lines, but real tracks can curve and end. It is inaccurate to describe a whole railway as two straight lines that run forever. A physical example is a model of part of the geometry, not the complete mathematical object.

Check and explain

Draw two parallel lines using a straightedge and a suitable guide, then turn the paper. Explain why the relationship stays the same. Next draw two short, nonparallel pieces whose full lines meet outside those pieces. Extend them to test your prediction.

Try the Geometry Sketchbook to explore endpoints, continuation, intersections and your own geometric drawings.