An angle is a figure formed by two lines or rays diverging from a common point. Two lines or rays, called the sides of the angle, and common point, called the vertex of the angle.
Angles
Suppose that the line $\overleftrightarrow{OP}$ is capable of revolving about its end point O. And, suppose that in this way it has passed successively from the position of the ray $\overrightarrow{OA}$ to the positions occupied by the rays $\overrightarrow{OB}$, $\overrightarrow{OC}$, $\overrightarrow{OD}$,… Then, the angle between the $\overrightarrow{OA}$ and any position such as $\overrightarrow{OB}$ is measured by the amount of revolution which the revolving line $\overleftrightarrow{OP}$ has undergone in passing between the position $\overrightarrow{OA}$ and $\overrightarrow{OB}$. Here, the revolving line $\overleftrightarrow{OP}$ is called the generating line.

In Euclidean Geometry, the direction in which the generating line revolves does not matter. Thus, the angle between the positions $\overrightarrow{OA}$ and $\overrightarrow{OB}$ can be denoted either by $\angle$AOB or by $\angle$BOA.

Identifying Angles

But in Trigonometry, the direction in which the revolving ray rotates does matter.

The angle $\angle$AOB denotes the amount of revolution which the revolving ray $\overrightarrow{OP}$ has undergone in passing from its initial position $\overrightarrow{OA}$ into its final position $\overrightarrow{OB}$ in the indicated direction (clockwise or anti-clockwise). The point O is called the origin, $\overrightarrow{OA}$ the initial vector, $\overrightarrow{OB}$ the terminating vector, and $\overrightarrow{OP}$ the radius vector.

Identifying Angle

Similarly, for the angle $\angle$BOA, the ray $\overrightarrow{OB}$ is the initial vector and the ray $\overrightarrow{OA}$ is the terminating vector. It is to be noted that the angle is positive if the radius vector revolves in anti-clockwise direction; and the angle is negative if the radius vector revolves in clockwise direction. The radius vector can make any number of revolutions in either clockwise or anti-clockwise direction. Thus, the angle can by any real number.

An angle is usually represented by a single letter. Different letters A, B, C,…, $\alpha$, $\beta$, $\gamma$,…, $\theta$, $\phi$, $\psi$,… are used to distinguish different letters.
The angles can be classified into the following:

Straight Angle: An angle whose sides lie in opposite directions from the vertex in the same straight line and whose measure is exactly 180.

Straight Angle

Let $\overleftrightarrow{A' A}$ be a straight line and O be a point between A and A'. Let $\overrightarrow{OP}$ be a radius vector with end point O. Then, the amount of revolution which the revolving ray $\overrightarrow{OP}$ has undergone in passing from its initial position $\overrightarrow{OA}$ into its final position $\overrightarrow{OA}$ is called a straight angle.

Right Angle: An angle $\angle$AOB is said to be a right angle, if the amount of revolution that the radius vector $\overrightarrow{OP}$ has undergone in passing from the position $\overrightarrow{OA}$ into the position $\overrightarrow{OB}$ is equal to half of the amount of revolution that the radius vector $\overrightarrow{OP}$ has undergone in making the straight angle $\angle AOA^'$.

Right Angle

It is clear that $\angle AOB \cong \angle BOA'$. Moreover, $\angle$AOB and $\angle$BOA' are supplementary angles.

Acute Angle: If the amount of revolution that the radius vector has undergoes in making an angle is less than the amount of revolution that it undergoes in making a right angle, then the angle is said to be an acute angle.

Obtuse Angle: If the amount of revolution that the radius vector has undergoes in making an angle is greater than the amount of revolution that it undergoes in making a right angle, then the angle is said to be an obtuse angle.

Right Acute Obtuse

In the figure, $\angle$AOB is a right angle, $\angle$AOC is an acute angle, $\angle$AOD is an obtuse angle.
It is to be noted that in Euclidean Geometry, an angle is always positive and is less than or equal to a straight angle.
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In order to measure angles, we must first select some fixed unit. The natural unit would be a right angle. This, however, is an inconvenient unit of measurement on account of its size. Thus, for convenience two systems of measurement have been established, in each of which the unit is a certain fraction of a right angle. The two systems of measurement are:

Sexagesimal Measure (English System): In this system of measurement, a right angle is divided into 90 equal parts called Degrees; a degree is divided into 60 equal parts called Minutes, and a minute into 60 equal parts called Seconds.

The symbols $1^o$, 1', 1" are used to denote a degree, a minute, and a second respectively. Thus,
60" = 1'
60' = $1^o$
$90^o$ = 1 Right Angle
Examples:
$91^o$ 20' 58", $18^o$ 21', and $100^o$ 0' 28"

Centesimal Measure (French System): In this system of measurement, a right angle is divided into 100 equal parts called Grades, a grade is divided into 100 equal parts called Minutes, a minute into 100 equal parts called Seconds.
The symbols $1^g$, 1', 1" are used to denote a degree, a minute, and a second respectively. Thus,
100" = 1'
100' = $1^g$
$100^g$ = 1 Right Angle
Examples:
$100^g$ 25' 80", $105^g$ 0' 92", and $108^g$ 22'

Even though the same names are used, a centesimal minute and second are not the same as a sexagesimal minute and second. Thus, a right angle contains 90×60=5400 sexagesimal minutes, whereas it contains 100 × 100 = 10000 centesimal minutes.

Circular Measure (Radian Measure): This is the third system of measurement which is using in all the higher branches of Mathematics. The unit used in this system of measurement is a radian, denoted by $1^c$, and is defined as the angle subtended at the center of any circle by an arc equal in length to the radius of the circle.

Circular Measure

Consider a circle with center O and radius r units. Let $\widehat{AB}$ be an arc with length r units. Then, the angle subtended by the arc $\widehat{AB}$ at the center O of the circle is a radian. It is to be noted that a complete revolution makes two straight angles and the circumference of the circle is $2 \pi r$ units. Thus, we have
$$\frac{2 \pi r}{r} = \frac{2 Straight\ Angles}{1 Radian}$$
$\rightarrow$ 1 Straight Angle = $\pi ^c$

It is customary to denote the angles in the circular measure by the small letters $\alpha, \beta, \gamma$,… of the Greek Alphabet.

The angles in these three systems of measurement can be converted from one into another by the following relationship:
1 Straight Angle = $180^o$ = $200^g$ = $\pi ^c$

It should be noted that
1 Right Angle =$90^o$ = $100^g$ = $\frac{\pi ^c}{2}$

Notation:
If $\angle$AOB is an angle, it is customary to denote its measure by m $\angle$AOB.
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Supplementary Angles: Two angles that add up to a straight angle are called supplementary angles. In the above figure, we can identify the following pairs of supplementary angles.

Angle Relationships

$\angle$APC and $\angle$CPB
$\angle$CPB and $\angle$BPD
$\angle$BPD and $\angle$DPA
$\angle$DPA and $\angle$APC
It is to be noted that supplementary angles need not be adjacent and the sum of the measures of supplementary angles is $180^o$ or $\pi ^c$.

Complementary Angles: Two angles that add up to a right angle are called complementary angles. Thus, the sum of the measures of complementary angles is $90^o$ or $\frac{\pi ^c}{2}$.

Congruent Angles: Two angles are said to be congruent, if they have the same measure.

Adjacent Angles: Two angles are said to be adjacent if they have a common vertex and a common side between them.
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A transversal is a line in a plane that intersects two (or more) other lines, which are on the same plane, at distinct points.

Transversal Angles

In the figure, the line t is intersecting two coplanar lines $l_1$ and $l_2$. The line t is called the transversal of the lines $l_1$ and $l_2$. Here, the lines $l_1$ and $l_2$ may or may not be parallel.

Transversal angles are the angles formed by a transversal and the lines cut by the transversal. If a transversal intersects two coplanar lines, it forms four angles at the intersection of each line and the transversal, as shown in the figure. These eight angles are called the transversal angles.
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