Perpendicular

Fig. 1: The line AB is perpendicular to the line CD, because the two angles it creates (indicated in orange and blue, respectively) are each 90 degrees.

For other uses, see Perpendicular (disambiguation).

In geometry, two lines or planes (or a line and a plane), are considered perpendicular (or orthogonal) to each other if they form congruent adjacent angles. The term may be used as a noun or adjective. Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B. Note that by definition, a line is infinitely long, and strictly speaking AB and CD in this example represent line segments of two infinitely long lines. Hence the line segment AB does not have to intersect line segment CD to be considered perpendicular lines, because if the line segments are extended out to infinity, they would still form congruent adjacent angles.

If a line is bending to another as in Figure 1, all of the angles created by their intersection are called right angles (right angles measure ½π radians, or 90°). Conversely, any lines that meet to form right angles are perpendicular.

In a coordinate plane, perpendicular lines have opposite reciprocal slopes. A horizontal line has slope equal to zero while the slope of a vertical line is described as undefined or sometimes ±infinity.

Numerical criteria

In terms of slopes

In a Cartesian coordinate system, two straight lines $L$ and $M$ may be described by equations.

$L :\; y\; =\; ax\; +\; b,$

$M :\; y\; =\; cx\; +\; d,$

as long as neither is vertical. Then $a$ and $c$ are the slopes of the two lines. The lines $L$ and $M$ are perpendicular if and only if the product of their slopes is -1, or if $ac=-1$.

The perpendiculars to vertical lines are always horizontal lines, and the perpendiculars to horizontal lines are always vertical lines. All horizontal lines are perpendicular to all vertical lines; that is, for any horizontal line $P :\; x\; =\; J$ and horizontal line $Q :\; y\; =\; K$, where $J$ and $K$ are constants, $P\; \backslash perp\; Q$.

Numerical criteria

In terms of slopes

In a Cartesian coordinate system, two straight lines $L$ and $M$ may be described by equations.

$L :\; y\; =\; ax\; +\; b,$

$M :\; y\; =\; cx\; +\; d,$

as long as neither is vertical. Then $a$ and $c$ are the slopes of the two lines. The lines $L$ and $M$ are perpendicular if and only if the product of their slopes is -1, or if $ac=-1$.

The perpendiculars to vertical lines are always horizontal lines, and the perpendiculars to horizontal lines are always vertical lines. All horizontal lines are perpendicular to all vertical lines; that is, for any horizontal line. One of the lines does NOT need to be upright $P :\; x\; =\; J$ and horizontal line $Q :\; y\; =\; K$, where $J$ and $K$ are constants, $P\; \backslash perp\; Q$.

In relationship to parallel lines

Fig. 3: Lines a and b are parallel, as shown by the tick marks, and are cut by the transversal line c.

As shown in Figure 3, if two lines (a and b) are both perpendicular to a third line (c), all of the angles formed on the third line are right angles. Therefore, in Euclidean geometry, any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate. Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.

In Figure 3, all of the orange-shaded angles are congruent to each other and all of the green-shaded angles are congruent to each other, because vertical angles are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent. Therefore, if lines a and b are parallel, any of the following conclusions leads to all of the others:

• One of the angles in the diagram is a right angle.
• One of the orange-shaded angles is congruent to one of the green-shaded angles.
• Line \'c\' is perpendicular to line \'a\'.
• Line \'c\' is perpendicular to line \'b\'.

See also

• Orthogonality
• Perpendicular component (of a vector)
• Surface normal
• Parallel (geometry)

External links

• Definition: perpendicular With interactive animation
• How to draw a perpendicular bisector of a line with compass and straight edge Animated demonstration
• How to draw a perpendicular at the endpoint of a ray with compass and straight edge Animated demonstration

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