Types of Angles in Maths: Names, Definition & Examples

Types of Angles: An angle is formed when two rays or lines intersect at the same point. Angles are a fundamental unit of geometry found in nature. It is used as an important aspect of architecture and engineering. Moreover, we use angles to measure changes in the trajectory of motion of ships, aeroplanes, stars, etc.

There are various types of angles in Maths. Some of the angles with names are Zero angles, Acute angles, Obtuse angles, Right angles, Reflex angles, and Straight angles. Let us learn about different types of angle forms, their properties, and how to measure them. Read on to understand the in-depth concept of angles, types of angles, and their properties, and learn important definitions and formulas.

What are Angles?

An angle is formed when two rays originate from the same originating point. Such rays make an angle which is called the arms of the angle and their originating point is called the vertex of the angle.

An angle that is represented by the symbol \(\angle \). Here from the above diagram, the formed angle is represented by (\angle PQR\). The same angle can also be represented as \(\angle RQP\). To angle is measured by degree.

The above two rays can combine in multiple ways to form the different types of angles in geometry. Let us begin by studying these different types of angles.

Types of Angles in Maths

The Angles can be classified into two main types:

1. Based on Magnitude
2. Based on Rotation

Types of Angles Based on Magnitude

There are \(7\) angle forms based on the magnitude or measurements of an angle. These are listed below:

1. Zero Angles
2. Acute Angle
3. Right Angles
4. Obtuse Angle
5. Straight Angle
6. Reflex Angle
7. Complete/Full Angle

Zero Angle

A zero angle \(\left({0^\circ } \right)\) is an angle formed when both the arms of angles are at the same position.

\(PQ\) and \(PR\) are two rays originating from the same point and are in the same direction. So, they represent zero angles.

Acute Angle

An acute angle is a type of angle whose measure is greater than \(0^\circ \) and less than \(90^\circ \). The figure below illustrates an acute angle.

Examples: \(30^\circ ,\,45^\circ ,\,60^\circ \) etc.

Right Angle

A right angle is a type of angle whose measure is exactly equal to \(90^\circ \). The figure below illustrates a right angle.

Obtuse Angle

An obtuse angle is a type of angle whose measure is greater than \(90^\circ \) and less than \(180^\circ \). The figure below illustrates an obtuse angle.

Examples: \(100^\circ ,\,120^\circ ,\,150^\circ \) etc.

Straight Angle

A straight angle is a type of angle whose measure is exactly equal to \(180^\circ \). It is also called \(180^\circ \) angle.

The name of a 180-degree angle is a straight angle. The figure below illustrates a straight angle. We can see, that it is just a straight line because the angle between its arms is (180^\circ ).

Reflex Angle

The definition of a reflex angle is that it is a type of angle whose measure is greater than \(180^\circ \) and less than \(360^\circ \). The figure below illustrates a reflex angle.

Example: \(200^\circ ,\,220^\circ ,\,250^\circ \) etc.

Complete/Full Angle

A complete angle is a type of angle whose measure is exactly equal to \(360^\circ \). It is also called a full angle. The figure below illustrates a complete/full angle.

Types of Angles Based on Rotation

Based on the direction of measurement or the direction of rotation, there are two angle forms:

1. Positive Angles
2. Negative Angles

Positive Angles

The angles that are measured by the counter-clockwise (or anti-clockwise) direction are called positive angles.

Negative Angles

The angles that are measured in a clockwise direction from the base are called negative angles.

Types of Angles Pairs

When 2 angles are paired, they form different angles, like:

1. Complementary angles
2. Supplementary angles
3. Linear pair
4. Adjacent angles
5. Vertically opposite angles

Complementary Angles

Two angles whose sum is of \(90^\circ \) are known as complementary angles. Whenever two angles are said to be complementary, each of the angles is said to be the complement of the other. As shown in the diagram below, \(30^\circ \) angle is the complement of \(60^\circ \) angle and vice versa because their sum is \(90^\circ \).

Supplementary Angles

Two angles whose sum is of \(180^\circ \) are known as supplementary angles. When two angles are supplementary, each angle is said to be the supplement of the other. As shown in the diagram below, \(60^\circ \) angle is the supplement of \(120^\circ \) angle and vice versa because their sum is \(180^\circ \).

Linear Pair

A linear pair of angles are formed when \(2\) lines intersect. Two angles are said to be linear if they are adjacent angles formed by \(2\) intersecting lines. The measure of a straight angle is \(180^\circ \). So, a linear pair of angles must be added up to \(180^\circ \).

In the below figure \(\angle 1\) and \(\angle 2\) are called the linear pair of angles.

Difference Between Pairs of Supplementary Angles and Linear Pairs of Angles

The pair of supplementary angles may or may not have a common vertex, but the linear pair of angles must always have a common vertex.

Adjacent Angles

Adjacent angles are two angles that have a common vertex and a common side but do not overlap. In the below figure, \(\angle 1\) and \(\angle 2\) are adjacent angles. They share the same vertex and the same common side.

Vertically Opposite Angles

When two lines intersect, the opposite angles are equal. In the diagram below, the angles \(\angle 1\) and \(\angle 2\) are equal. These angles \(\angle 1\) and \(\angle 2\) are called vertically opposite angles because they are opposite to each other at the vertex.

Angles Formed by Transversal

A line that intersects two or more lines at different points is known as a transversal. The angles formed at the point of intersection are:

1. Interior Angles: The inner of the two angles formed where two sides of a polygon come together. \(\angle 3,\,\angle 4,\,\angle 5\) and \(\angle 6\) are interior angles.
2. Exterior Angles: The exterior Angle is the angle between any side of a shape, and a line extended from the next side. \(\angle 1,\angle 2,\,\angle 7\) and \(\angle 8\) are exterior angles.
3. Pairs of Alternate Interior Angles: Pairs of alternate interior angles are pair angles formed when a line intersects two parallel lines. Alternate interior angles are always equal. \(\angle 3\,\& \,\angle 6\) and \(\angle 4\,\& \,\angle 5\) are two pairs of alternate interior angles. \(\angle 3 = \angle 6\) and \(\angle 4 = \angle 5\)
4. Pairs of Alternate Exterior Angles: Pairs of alternate exterior angles are simply vertical angles of the alternate interior angles. Alternate exterior angles are equal. \(\angle 1\,\& \,\angle 8\) and \(\angle 2\,\& \,\angle 7\) are two pairs of alternate exterior angles. \(\angle 1 = \angle 8\) and \(\angle 2 = \angle 7\)
5. Pairs of Corresponding Angles: Corresponding angles are pair angles formed when a line intersects a pair of parallel lines. Corresponding angles are also equal. \(\angle 1 = \angle 5,\,\angle 3 = \angle 7,\,\angle 2 = \angle 6\) and \(\angle 4 = \angle 8\) are four pairs of corresponding angles
6. Pairs of Interior Angles on the Same Side of the Transversal: \(\angle 3\,\& \,\angle 5\) and \(\angle 4\,\& \,\angle 6\) are two pairs of interior angles on the same side of the transversal. These pairs of angles are supplementary. So, \(\angle 3 + \angle 5 = 180^\circ \) and \(\angle 4 + \angle 6 = 180^\circ \).

Properties of Angles

The properties of angles are:

a) Zero angle measures exactly \(0^\circ \).
b) Acute angle measures more than \(0^\circ \) and less than \(90^\circ \).
c) Right angle measures exactly \(90^\circ \).
d) Obtuse angle measures more than \(90^\circ \) and less than \(180^\circ \).
e) Straight angle measures exactly \(180^\circ \).
f) Reflex angle measures more than \(180^\circ \) and less than \(360^\circ \).
g) Complete angle measures exactly \(360^\circ \).
h) The sum of all the angles on one side of a straight line always measures \(180^\circ \).
i) The sum of all the angles around a point always measures \(360^\circ \).

Types of Angles Examples

Below we have provided some examples of various types of angles.

a) Example of Zero Angle

The angle between the hour and minute hand as shown in the clock above forms a zero angle.

b) Example of Acute Angle

The angle between the hour and minute hand as shown in the clock above forms an acute angle.

c) Example of Right Angle

The angle between the hour and minute hand as shown in the clock above forms the right angle.

d) Example of Obtuse Angle

The angle between the hour and the minute hand as shown in the clock above forms an obtuse angle.

e) Example of Straight Angle

The angle between the hour and minute hand as shown in the clock above forms a straight angle.

f) Example of Reflex Angle

The angle between the hour and minute hand as shown in the clock above forms a reflex angle.

g) Example of Complete Angle

The angle between the hour and minute hand as shown in the clock above forms a complete angle.

Practice Questions on Angles

Q.1: Find the value of angle \(\angle C\) for the given triangle \(ABC\), where \(\angle A = 25^\circ \) and \(\angle B = 90^\circ \).
Ans:
Given, \(\angle A = 25^\circ \) and \(\angle B = 90^\circ \)
Here, we need to find the angle \(\angle C\).
We know that,
The sum of interior angles of a triangle is \(180^\circ \).
Thus, \(\angle A + \angle B + \angle C = 180^\circ \)
\( \Rightarrow 25^\circ + 90^\circ + \angle C = 180^\circ \)
\( \Rightarrow 115^\circ + \angle C = 180^\circ \)
\(\Rightarrow \angle C=180^{\circ}-115^{\circ}\)
\( \Rightarrow \angle C = 65^\circ \)
Hence, the value of the angle \(\angle C\) is \(65^\circ \).

Q.2: Find the value of angle \(\angle D\) in a quadrilateral \(ABCD\), where \(\angle A=100^{\circ}, \angle B=60^{\circ}, \angle C=35^{\circ}\).
Ans:
\(\angle A = 100^\circ \)
\(\angle B = 60^\circ \)
\(\angle C = 35^\circ \)
Here, we need to find the value of \(\angle D\).
We know that the sum of the interior angles of a quadrilateral is equal to \(360^\circ \).
Therefore, 
\(\angle A + \angle B + \angle C + \angle D = 360^\circ \ldots .(1)\)
Now, substitute the known values in the equation \((1)\), we get,
\(100^\circ + 60^\circ + 35^\circ + \angle D = 360^\circ \) 
\( \Rightarrow 195^\circ + \angle D = 360^\circ \)
\( \Rightarrow \angle D = 360^\circ – 195^\circ \)
\( \Rightarrow \angle D = 165^\circ \)
Hence, the value of the angle, \(\angle D\) is \(165^\circ \).

Q.3: Identify the type of angle by observing the below figure.

Ans:

Given,

The measurement of the given angle \(=185^\circ \).
We know that the angle measured between \(180^\circ \) and \(360^\circ \) is called a reflex angle.
The given angle i.e. \(185^\circ \) comes in between \(180^\circ \) and \(360^\circ \).
Hence, the angle in the given figure is a reflex angle.

Q.4: Find the angle complementary to the angle \(50^\circ \).
Ans: 
The given angle is \(50^\circ \).
Here, we need to find the complementary of it.
We know that two angles are complementary when they add up to \(90^\circ \).
Let the complementary angle be \(x\).
Now, \(x + 50^\circ = 90^\circ \)
\(x = 90^\circ – 50^\circ = 40^\circ \)
Hence, the complementary angle is \(40^\circ \).

Q.5: Find the supplementary of the angle \(120^\circ \).
Ans: 
The given angle is \(120^\circ \).
Here, we need to find a supplementary of it.
We know that two angles are supplementary when they add up to \(180^\circ \).
Let the supplementary angle be \(x\).
Now, \(x + 120^\circ = 180^\circ \)
\(x = 180^\circ – 120^\circ = 60^\circ \)
Hence, the supplementary angle is \(60^\circ \).

Summary

We have learned that an angle is a geometric figure formed using two rays through this article. The types of angles like zero, acute, right, obtuse, reflex, straight, complete, positive and negative angles, complementary angles, supplementary angles, transversal angles, etc., have been discussed here. Angles that are formed by a transversal are corresponding, alternate interior angles Alternate exterior angles, co-interior angles, co-exterior angles and vertically opposite angles have also been discussed.

FAQs on Types of Angles

Q.1: How do you describe angles?
Ans: An angle can be described as the figure formed by two rays meeting at a common endpoint. An angle is represented by the symbol \(\angle .\) Angles are measured in degrees using the protractor.

Q.2: What are an angle and its types?
Ans: An angle is formed when two rays meet at a common endpoint. The rays making an angle are called the arms of an angle and the common endpoint is called the vertex of an angle. There are \(7\) types of angles in Maths. These are zero angles, acute angles, right angles, obtuse angles, straight angles, reflex angles, and complete angles.

Q.3: Which kind of angle is the largest?
Ans: A complete angle is a type of angle whose measure is exactly equal to \(360^\circ \). It is also called a full angle. So, the complete angle is the largest.

Q.4: How do you identify angles?
Ans: We can identify an angle by visualising the figure or by measuring the angles through a protractor.

Q.5: What is a vertical angle?
Ans: Angles that have a common vertex, and the sides of the angle are formed by the same lines are called vertical angles. Vertical angles are equal.

Q.6: What is the name of a 180-degree angle?
Ans: The 180-degree angle name is the straight angle.

Q.7: What is the reflex angle definition?
Ans: According to the definition, a reflex angle is an angle whose measure is greater than (180^\circ ) and less than (360^\circ ).

Students can also make use of Class 6 Maths Practical Geometry NCERT Solutions which will be helpful in their preparation.

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