# Definition:Right Angle/Perpendicular

## Definition

In the words of Euclid:

*When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is***right**, and the straight line standing on the other is called a**perpendicular**to that on which it stands.

(*The Elements*: Book $\text{I}$: Definition $10$)

In the above diagram, the line $CD$ has been constructed so as to be a **perpendicular** to the line $AB$.

### Foot of Perpendicular

The **foot** of a perpendicular is the point where it intersects the line to which it is at right angles.

In the above diagram, the point $C$ is the **foot** of the perpendicular $CD$.

## Line Perpendicular to Plane

In the words of Euclid:

*A***straight line**is**at right angles to a plane**when it makes right angles with all the straight lines which meet it and are in the plane.

(*The Elements*: Book $\text{XI}$: Definition $3$)

In the above diagram, the line $AB$ has been constructed so as to be a **perpendicular** to the plane containing the straight lines $CD$ and $EF$.

## Plane Perpendicular to Plane

In the words of Euclid:

*A***plane**is**at right angles to a plane**when the straight lines drawn, in one of the planes, at right angles to the common section of the planes are at right angles to the remaining plane.

(*The Elements*: Book $\text{XI}$: Definition $4$)

In the above diagram, the two planes have been constructed so as to make lines perpendicular to their common section perpendicular to each other.

Thus the two planes are perpendicular to each other.

## Also known as

The word **normal** is often used for **perpendicular**, particularly in the context of **vector analysis**.

Also, in the context of **linear algebra** and **analysis**, the word **orthogonal** is often encountered, which is a generalization of the concept of **perpendicularity**, but in a more abstract context than **geometry**

## Also see

- Results about
**perpendiculars**can be found**here**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**perpendicular** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**perpendicular**