Perpendicular lines

When 2 lines intersect at a right angle, they are perpendicular lines. We can also say that if 2 lines are perpendicular, then their intersection forms a right angle. Sometimes in everyday language, parts of lines (rays and line segments) that meet at right angles are also called perpendicular lines.

Lines l and m are perpendicular because they meet at a right angle. We can write l⊥m to show this, where "⊥" is the symbol for perpendicular.

Also, if two lines are perpendicular, 4 right angles are created. In the diagram below, angles 1, 2, 3, 4 are all right angles.

Properties of perpendicular lines

• There can be only one line that is perpendicular to a given line at a given point.

Let lines m, n, and l intersect at point P above. If the angle formed by l and m is a right angle, then the angle formed by lines l and n cannot also be a right angle.

• Two lines are perpendicular if their slopes are negative (or opposite) reciprocals of each other.

The slope of line p is ½ and the slope of line r is -2, the negative reciprocal of ½, so p and q are perpendicular and meet at a right angle.

Knowing that two perpendicular lines meet at a right angle, or that if their intersection forms a right angle that they are perpendicular, is useful information in working with postulates, theorems, and other properties in Geometry. Below are a few examples.

• If the two non-adjacent sides of two acute adjacent angles are perpendicular then, the angles must be complementary.

∠BAD and ∠CAD are adjacent angles, with non-adjacent, perpendicular sides AB and AC. Since AB and AC are perpendicular, ∠BAC is a right angle measuring 90° so, ∠BAD + ∠CAD = 90°.

• If a transversal is perpendicular to one of a pair of parallel lines (or several parallel lines), it must be perpendicular to all the parallel lines.

In the diagram above, m//n and l⊥m. Since l⊥m, all four angles formed by the intersection of l and m are right angles. Also, since m//n all four angles are also right angles formed by lines l and n, by the property of corresponding angles of parallel lines being congruent. Thus, line l is also perpendicular to line n.

Did you know?

The x- and y-axis are two lines in the Cartesian coordinate plane that are perpendicular to each other.

Also, each horizontal grid line is perpendicular to each vertical grid line that comprises the grid for the system. The Cartesian coordinate system is also known as an orthogonal (meaning at right angles) coordinate system.