# Definition:Angle

## Definition

Let $\LL_1$ and $\LL_2$ be two lines which intersect at a point $\PP$.

The angle between $\LL_1$ and $\LL_2$ is defined as the inclination between $\LL_1$ and $\LL_2$ at $\PP$.

In the words of Euclid:

A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.

### Rectilineal

In the words of Euclid:

And when the lines containing the angle are straight, the angle is called rectilineal.

Thus the distinction is made between straight-line angles and curved-line angles.

Most of the time the fact that angles are rectilineal is taken for granted.

### Arm

Each of the two intersecting lines or line segments that form an angle are called its arms.

### Subtend

Let $AB$ be a line segment and $C$ be a point:

The line segment $AB$ is said to subtend the angle $\angle ACB$.

Two angles are adjacent if they have an intersecting line in common:

### Containment

The two arms of an angle are said to contain that angle.

### Vertex

The point at which the arms of an angle meet is known as the vertex of that angle.

## Notation

In order to refer to an angle in an exposition, it is common to use one of the symbols:

$\angle ABC$ or $\widehat {ABC}$

to denote the angle formed by the intersection of the lines $AB$ and $BC$.

The $\LaTeX$ code for $\angle ABC$ is \angle ABC .

The $\LaTeX$ code for $\widehat {ABC}$ is \widehat {ABC} .

## Measurement

The usual units of measurement for angle are as follows:

### Degree

The degree (of angle) is a measurement of plane angles, symbolized by $\degrees$.

 $\ds$  $\ds 1$ degree $\ds$ $=$ $\ds 60$ minutes $\ds$ $=$ $\ds 60 \times 60 = 3600$ seconds $\ds$ $=$ $\ds \dfrac 1 {360}$ full angle (by definition)

### Minute

The minute (of angle) is a measurement of plane angles, symbolized by $'$.

 $\ds$  $\ds 1$ minute $\ds$ $=$ $\ds 60$ seconds $\ds$ $=$ $\ds \dfrac 1 {60}$ degree of angle (by definition)

### Second

The second (of angle) is a measurement of plane angles, symbolized by .

 $\ds$  $\ds 1$ second $\ds$ $=$ $\ds \dfrac 1 {60}$ minute of angle (by definition) $\ds$ $=$ $\ds \dfrac 1 {60 \times 60} = \dfrac 1 {3600}$ degree of angle

The radian is a measure of plane angles symbolized either by the word $\radians$ or without any unit.

Radians are pure numbers, as they are ratios of lengths. The addition of $\radians$ is merely for clarification.

$1 \radians$ is the angle subtended at the center of a circle by an arc whose length is equal to the radius:

## Types of Angle

Angles can be divided into categories:

### Zero Angle

The zero angle is an angle the measure of which is $0$ regardless of the unit of measurement.

### Acute Angle

An acute angle is an angle which has a measure between that of a right angle and that of a zero angle.

### Right Angle

A right angle is an angle that is equal to half of a straight angle.

### Obtuse Angle

An obtuse angle is an angle which has a measurement between those of a right angle and a straight angle.

### Straight Angle

A straight angle is defined to be the angle equal to one half of a complete turn.

### Reflex Angle

A reflex angle is an angle which has a measure between that of a straight angle and that of a full angle.

### Full Angle

A full angle is an angle equivalent to one full rotation.

It is possible to consider angles outside the range $\closedint {0 \degrees} {360 \degrees}$, that is, $\closedint 0 {2 \pi}$.

However, in geometric contexts it is usually preferable to convert these to angles inside this range by adding or subtracting multiples of a full angle.

## Directed versus Undirected Angles

The most basic definition of angle is an undirected angle on the interval $\closedint {0 \degrees} {180 \degrees}$ or $\closedint 0 \pi$.

This definition is often insufficient, in cases such as the external angles of a polygon.

Therefore, angles are most commonly defined in one of two ways:

$(1): \quad$ Undirected angles on the interval $\closedint {0 \degrees} {360 \degrees}$ or $\closedint 0 {2 \pi}$.
$(2): \quad$ Directed angles, with the positive direction being anticlockwise from a given line (or, if no line is specified, from the $x$-axis).
This definition is more commonly found in applied mathematics, such as in surveying, navigation, or, more colloquially, in a $720 \degrees$ degree spin in skateboarding, skiing, etc.

## Also see

• Results about angles can be found here.