## Angle Definition

Have you seen a kite? Apart from the vibrant colours in it, have you noticed that each side of it makes an angle with another part at the corner? From the image below, you can see that whenever two lines are intersecting, an angle is formed between the lines.

Learning about angles is essential as it forms the foundation of geometry and other mathematical concepts.

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You can find angles in many other things that you use in your everyday life like a pair of scissors, a chair, a hockey stick, etc.

In this article, we will take a close look at the definition of angle in maths, various concept of angles, and the different types of angle that exist in mathematics.

What is an Angle

The word angle finds its origin from a Latin word “Angulus” whose meaning is “a small bending” and the concept of angles was introduced by Eudemus (an ancient Greek philosopher) who gave the definition of an angle as a deviation from a straight line.

So, what is the definition of angle in mathematical terms? When two lines meet at a single shared point, they form an angle. The lines forming an angle don’t need to be only straight lines, an angle can also be formed by the intersection of curved lines. Another angle definition could be the degree of turn of a line around a central point.

To clarify the angle meaning better, let us look at various parts of an angle with the help of a diagram:

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Arm - The two rays that come together to form the angle are called arms of the angle or sides of the angle. So in the above diagram, OB and OA are the arms of the angle AOB.

Vertex - The endpoint, which is common to both the arms of an angle, is called the vertex of the angle. In the given figure, O is the vertex of angle AOB.

Angles are measured in degrees and represented by the symbol °.

## Types of Angles

There are various types of angles that are based on the measure of their degrees. With the increase in the angle, its name changes.

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### Acute Angle

If you look at the letter “A” in the word “Acute”, the angle between the lines is an acute angle. Any angle which is less than 90 degrees is an acute angle. Their value lies between 0 degrees and 90 degrees i.e. 0° < 𝛉 < 90°.

### Right Angle

An angle that is equal to 90 degrees is a right angle. It is mostly formed by two perpendicular lines and 𝛉 = 90°.

### Obtuse Angle

An angle that is greater than 90 degrees and less than 180 degrees is an obtuse angle. Its value lies between 90 degrees and 180 degrees i.e. 90° < 𝛉 < 180°.

In a straight line, Obtuse angle = 180 - acute angle.

### Straight Angle

A straight angle is exactly 180 degrees. Since it is similar to a straight line, that is why it is termed as the straight angle. A straight angle is a mixture of acute and obtuse angles on a straight line, i.e. Straight angle = Obtuse angle + acute angle.

### Reflex Angle

An angle that is greater than 180 degrees and less than 360 degrees is an obtuse angle. Its value lies between 180 degrees and 360 degrees i.e. 180° < 𝛉 < 360°. You can calculate the reflex angle if you know the value of the acute angle since it is the complementary angle to the acute angle on the other side of the straight line.

### Complete Angle

If you rotate a straight line so that it finally reaches its original starting position, it forms a complete angle that is equal to 360 degrees. This is referred to as the complete angle or a full angle. Hence 𝛉 = 360 degrees here. Another name for this angle is perigon, and it is the central angle of a circle.

Complete angle = four right angles = two straight angles.

## Relationships Between Angles

Angles are related to each other based on either their summation or where they lie on two intersecting lines:

### Complementary Angles

Two angles are called complementary angles to each other if they sum up to 90 degrees. It is not necessary for the angles to be adjacent to each other to be called complementary angles. As long as their summation is 90 degrees, they are termed complementary angles. In the diagram below, both sets of angles are complementary. In the first one, they are adjacent to each other while in the second one they are not.

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### Supplementary Angles

Supplementary angles are angles whose sum is 180 degrees. Supplementary angles could be of different types as described below:

Vertical angles - When two lines cross, the angles opposite to each other are called vertical angles. Vertical angles are equal in measure. In the figure below, angles 1 and 3 are vertical angles and equal to each other. Angles 2 and 4 also form vertical angles and are equal to each other.

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Alternate interior angles - When we draw a traversal to two straight lines, the angles on the opposite sides of the traversal that lie on the interior side constitute the alternate interior angles. If the two lines are parallel to each other, then alternate interior lines are equal. Angles 2 and 3 are alternate interior angles in the image below.

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Alternate exterior angles - When we draw a traversal to two straight lines, the angles on the opposite sides of the traversal that lie on the exterior side constitute the alternate exterior angles. If the two lines are parallel to each other, then alternate exterior lines are equal. Angles 1 and 4 are alternate exterior angles here.

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Corresponding angles - When a line crosses two lines, the angles in matching corners are called corresponding angles. One of them is internal, and another is external. Corresponding angles are equal if the two lines (which are intersected) are parallel to each other. In the image below angles, 1 and 2 are corresponding angles. Here 1 is the external angle, and 2 is the internal angle.

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Adjacent angles - Two angles with a common vertex and a side are called adjacent angles.

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Q1: What Kinds of Angles are Based on the Direction of Rotation?

Ans: Based on the direction of rotation, angles are divided into two categories:

Positive angle - In a cartesian plane, positive angles are drawn from the origin in a counterclockwise direction, and they lie in the (+x, +y) quadrant.

Negative angle - In a cartesian plane, negative angles are drawn from the origin in a clockwise direction, and they lie in the (+x, -y) quadrant.

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Q2: What are Interior and Exterior Angles?

Ans: In a closed figure, the angle which lies within the boundaries of the shape are called interior angles while the angles that are formed outside the shape with any one side of the shape as a common side (by extending the line from the adjoining side) are called exterior angles. In the image below angle, ACB is an internal angle while angle ACD is an exterior angle:

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Q3: Explain the Relationship Between Degrees and Radians?

Ans: Angles can be measured both in degrees and radians. The radian unit of measurement is based on the radius of a circle. The circumference of a circle is 2π * r (radius) which means that a circle is approximately 6.28 radians.

So 2π * r = 360 degrees hence 1 radian = 180/π

Similarly 1 degree = π/180 radians