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 (a T-shape). 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. Two lines that are perpendicular would be denoted as ABCD

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

 :

 :

as long as neither is vertical. Then and are the slopes of the two lines. The lines and are perpendicular if and only if the product of their slopes is -1, or if .

To make the perpendicular to the line AB through the point P using compass and straightedge, proceed as follows (see Figure 2).

• Step 1 (red): construct a circle with center at P to create points A' and B' on the line AB, which are equidistant from P.
• Step 2 (green): construct circles centered at A' and B', both passing through P. Let Q be the other point of intersection of these two circles.
• Step 3 (blue): connect P and Q to construct the desired perpendicular PQ.

To prove that the PQ is perpendicular to AB, use the SSS congruence theorem for triangles QPA' and QPB' to conclude that angles OPA' and OPB' are equal. Then use the SAS congruence theorem for triangles OPA' and OPB' to conclude that angles POA and POB are equal.

As shown in Figure 3, if two lines (a and b) are both perpendicular to a third line (c), all of the angles formed along 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'.

In algebra, for any linear equation y=mx + b, the perpendiculars will all have a slope of (-1/m), the opposite reciprocal of the original slope. It is helpful to memorize the slogan "to find the slope of a perpendicular line, flip the fraction and swap the sign." Recall that any integer a is itself over one, and can be written as (a/1)

To find the perpendicular of a given line which also passes through a particular point (x, y), solve the equation y = (-1/m)x + b, substituting in the known values of m, x, and y to solve for b.

The perpendicular symbol is . For example, indicates that line AB is perpendicular to line CD.

In the Unicode character set, the perpendicular sign has the codepoint U+27C2 and is part of the Miscellaneous Mathematical Symbols-A range. It often looks the same as the "up tack" symbol (U+22A5), but is a different traits.

Perpendicular can be found in certain 108 piece Crayola crayon packs. The RGB of Perpendicular R value of 46 a G value of 63 and a B value of 65. This shade of purple is a common color that is produced when a projector is malfunctioning.

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

• 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