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.
(The Elements: Book $\text{I}$: Definition $8$)
Rectilineal
In the words of Euclid:
- And when the lines containing the angle are straight, the angle is called rectilineal.
(The Elements: Book $\text{I}$: Definition $9$)
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 $\angle ACB$.
Adjacent
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 |
Radian
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.
Sources
- 1947: William H. McCrea: Analytical Geometry of Three Dimensions (2nd ed.) ... (previous) ... (next): Chapter $\text {I}$: Coordinate System: Directions: $\S 1$. Introductory: Metrical geometry
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): angle
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): angle
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): angle (between lines in space)