Perpendicular

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For other uses, see Perpendicular (disambiguation).

In geometry, two lines are considered perpendicular if one falls on the other in such a way as to create 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.

If a line is perpendicular to another as in Figure 1, all of the angles created by their intersection are called right angles (right angles measure <math>\frac{\pi}{2}</math> radians, or 90°). Conversely, any lines that meet to form right angles are perpendicular. The line AB does not have to end at B to be considered perpendicular.

1 Numerical criteria
- 1.1 In terms of slopes
2 Construction of the perpendicular
3 In relationship to parallel lines
4 See also
5 External links

[edit] Numerical criteria

[edit] In terms of slopes

In a Cartesian coordinate system, two straight lines <math>L</math> and <math>M</math> may be described by equations

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

[edit] Construction of the perpendicular

Image:Perpendicular-construction.svg

To construct 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.

[edit] In relationship to parallel lines

Image:Perpendicular transversal v3.svg

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 a 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 and alternate interior angles are congruent. Therefore, if lines a and b are parallel, any of the following conclusions leads to all of the others: