Angles | Definition, Types and Examples

In geometry, an angle is a figure that is formed by two intersecting rays or line segments that share a common endpoint. The word “angle” is derived from the Latin word “angulus”, which means “corner”. The two lines joined together are called the arms of the angle and the measure of the opening between them is the value of the angle between these two lines.

In this article, we will learn about the angles their definition, and their parts in Geometry, their representation, examples along with their types like acute angle, right angle, obtuse angle, etc.

Angle Definition

 An Angle is a shape or space formed at the meeting point of two intersecting rays or line segments.

An angle is formed when two rays or line segments are joined together at a common point. The two lines are called 'Arms of the Angle' and the common point of the meeting is called a 'Vertex'.

Symbol of Angle

Angle is represented by the symbol "∠".

Representation of an Angle

While writing an angle "∠" is used along with the points. The common point is written in between the two other points. For Example, if we have ∠AOB, it means O is the common point at which two rays OA and OB are meeting and an angle is formed.

Check: Lines and Angles

Parts of an Angle

An angle consists of the following parts :

• Arms
• Vertex
• Initial Side
• Terminal Side

The image shown below shows the parts of an angle.

Arms of an Angle

The two rays that joint together to form the angles are called the arms of the angle. In the image added below, OP and OQ are the arms of the angle. The space between these two arms is the measure of the angle.

Vertex of an angle

The endpoint of the two arms of the angles is called the vertex of the angle. It is the point where the arms of the angle meet. In the above image added O is the vertex of the angle.

Types of Angles

There are different types of anglebased on basis of four different parameters. They are mentioned as follows:

There are six types of angles on the basis of measurement. They are:

• Acute Angle
• Right Angle 
• Obtuse Angle
• Straight Angle
• Reflex Angle
• Complete Angle

Acute Angle

The angle that measures less than 90° is called as Acute Angle. The degree always measures between 0 and 90. Acute angles measure positive when the rotation is anticlockwise and negative when the rotation of the angle is clockwise.

In the figure, O is the vertex of the angle, and OA and OB are two intersecting rays that meet at point O forming an angle less than 90°. Hence, ∠AOB is an acute angle.

Right Angle

The angle that exactly measures 90° is called a Right Angle. It is also considered as a half straight angle as half of 180° makes a right angle. The value of the angle may be positive or negative on the basis of the rotation of the angle.

In the figure, O is the vertex of the angle, and OA and OB are two intersecting rays that meet at point O forming an angle of exactly 90°. Hence, ∠AOB is a right angle. Also when two rays meet to form a 90° angle there are called Perpendicular to each other. Here OA and OB are perpendicular to each other.

Learn More, 7 Types of Angles

Obtuse Angle

The angle that measures more than 90° and less than 180° is called an Obtuse Angle. The degree always lies between 90° and 180°. The value of the obtuse angle will be positive if the rotation is anticlockwise and negative if the rotation is clockwise.

In the figure, O is the vertex of the angle, and OA and OB are two intersecting rays that meet at point O forming an angle of more than 90°. Hence, ∠AOB is an obtuse angle.

Straight Angle

The angle that measures exactly 180° is a Straight Angle. It is called Straight Angle because when two rays make 180° between them then they are in a straight line.

In the figure, of a straight angle, we can observe that O is the meeting point of two arms, called the vertex and OA and OB are two sides of the angle.

Reflex Angle

The angle that measures more than 180° and less than 360° is called a Reflex Angle. The degree always lies between 180° and 360°.

In the figure, O is the vertex of the angle, and OA and OB are two intersecting rays that meet at point O forming an angle of more than 180°. Hence, ∠AOB is a reflex angle.

Complete Angle

The angle whose measurement is 360° is called a Complete Angle. It happens when you make a complete turnaround and reach the initial point then in this case the angle is Complete Angle. 

Let's now study angle on the basis of Rotation

Positive and Negative Angles

There are two types of angles on the basis of Rotation. They are listed as follows:

• Positive Angle
• Negative Angle

Positive Angle

The angle that moves anticlockwise from its base and is drawn from the point (x, y) which is its origin is called a positive angle.

Negative Angle

 The angle that moves clockwise from its base and is drawn from the point (-x, -y)  which is its origin is a negative angle.

Now we will study the angles on the basis of pair.

Types of Angles in Pair

There are five types of angles on the basis of pairs. They are:

• Complementary Angles
• Supplementary Angles
• Adjacent Angles
• Linear Pair
• Vertically Opposite Angles

Complementary Angles

If the sum of two angles measures 90° then, the angles are said to be Complementary Angles and each angle is called a complement of the other.

The two angles combining together do not require to be adjacent or similar. It can be any two types of angles measuring 90° after addition. For Example, 70 and 20 are complementary angles.

Supplementary Angles

If the sum of two angles measures 180°, the angles are said to be Supplementary Angle. Each Angle is called a Supplement of the other.

For Example, 150° and 30° are Supplementary Angles.

Adjacent Angles

Two angles are said to be adjacent if they have a common vertex, a common arm, and the rest two arms lie on the alternate side of the common arm. Angle AOC and Angle BOC are Adjacent Angles

∠AOC and ∠BOC are here adjacent because they have a common point O, a common vertex OC and rest two arms OA and OB lie on the alternate side of the common arm.

Linear Pair

When the sum of two adjacent angles is 180° then they are called a Linear Pair.

As the name suggests the pair of angles result in a straight line.

Remember that there is one difference between Supplementary Angle and Linear Pair. For Linear Pair, the two angles must be adjacent while there is no such condition for Supplementary Angles. For Supplementary Angles, only the sum of the angles should be 180° doesn't matter if they are adjacent or not.

Here ∠AOC and ∠BOC are linear pairs as AOB is a straight line.

Vertically Opposite Angles.

When two lines intersect each other at a common point then the pair of angles in front of each other are called Vertically Opposite Angles.

• In the below figure, AB and CD are two lines that intersect each other at O, then pairs of Vertically Opposite Angles are (∠AOC, ∠BOD) and (∠AOD, ∠BOC).
• It should be noted that a pair of vertically opposite angles are equal i.e. ∠AOC = ∠BOD) and ∠AOD = ∠BOC).

Angles Formed by Transversal and Parallel Lines

There are four types of angles formed by transversal and parallel lines. They are :

• Corresponding Angles
• Alternate Interior Angles
• Consecutive Interior Angles
• Alternate Exterior Angles

Corresponding Angles

The Angles that are present at similar positions and on the same side of the transversal are Corresponding Angles. Corresponding Angles are the same in measurement.

In the figure below, ∠AOL and ∠CPM are corresponding angles placed at similar positions one at the exterior and the other at the interior part.

Alternate Interior Angles

The angles which are present on opposite sides of the transversal is the alternative interior angle. They are present at the inner side of the Z formed in the figure. The pair of Alternate Interior angles are equal to each other.

In the figure below, ∠AOT and ∠OTR are alternate interior angles placed interiorly alternate to each other. Similarly, ∠BOT and ∠OTQ are also Alternate Interior Angles.

Alternative Exterior Angles

The angles present on opposite sides of the transversal but externally are the alternative exterior angle. They are spotted at the exterior part of Z and both the angles measure the same.

In the figure below, (∠AOL, ∠DPM) and (∠BOL, ∠CPM) are the pair of Alternate Exterior Angles.

Consecutive Interior Angles

When two interior angles of the same side of the transversal are placed consecutively i.e. just after the other then they are called Consecutive Interior Angles. The sum of the pair of Consecutive Interior Angles is 180°.

In the below figure pair of Consecutive Interior Angles are (∠BOP, ∠CPO) and (∠AOP, ∠OPD).

Interior and Exterior Angles

Interior and exterior angles depends on the region of the angle where they are made. Let's learn about interior and exterior angles below.

Interior Angle

The angles that are formed inside any shape are called the Interior angles. For Example, angle inside a triangle, quadrilateral etc.

Exterior Angle

The angle that are formed outside any shape are called the exterior angles. Suppose we take a triangle ABC then and extend the line BC to D then in that figure we can easily mark the interior and exterior angles.

Here, in the above figure, ∠ABC, ∠BCA, and ∠CAB are interior angles and ∠ACD is the exterior angle.

Measuring an Angle

• Angle can be measured in 'Degree' or 'Radian'. In the case of Degrees, the measurement goes from 0° to 360° while in the case of Radian measurement goes from 0 to π.

• Smaller units of angle are minutes and seconds. Minute is represented by a single apostrophe(') while second is represented by a double apostrophe('').

We should remember below mentioned relations among various units of angles:

• π = 180°
• 1° = 60'
• 1' = 60''

Degree of an Angle

To convert the Angle from Degree to Median, we should multiply the given angle(in degrees) by π/180. Let's see one example

Example: Convert 90° to Radian

Solution:

90° × π/180 = π/2

Radian of an Angle

To convert the Angle from Radian to Degree we should multiply the given angle(in radians) by 180/π. Let's see one example

Example: Convert π/2 to Degrees.

Solution:

(π/2) × (180/π) = 90°

Learn more, Degrees to Radians

How to Measure an Angle?

An angle can be measured easily by using a protector or compass. In general, we use the protector to measure the angles. Follow the steps added below to measure the angle,

• Step 1: Place the protector above one of the arms of the angle.

• Step 2: Measure the value in the anticlockwise direction or clockwise direction depending on the opening of the angle then mark the value where the angle's arms coincide with the value in the protector.

• Step 3: The reading obtained in the protector is the required measure of the angle.

Steps to Construct an Angle

An angle can be easily constructed using the proctor or compass. To construct an angle using the protractor we follow the steps added below,

Step 1: Draw a ray OA of any length that is parallel to horizontal edge of page.

Step 2: Place the protractor on the ray OA such that O is at the centre of the protractor. And OA is at the right side of the protractor.

Step 3: Mark the point from the right side of the protractor at the angle which we want to construct suppose we have to construct an angle of 60°(mark the point as P)

Step 4: Join OP ∠AOP is the required angle.

Article Related to Angles:

• Lines and Angles
• Angles Formula
• Types of Angles

Solved Examples on Angles

Example 1: Find the complementary angle of ∠A = 48^o.

Solution:

Given angle,

∠A = 48^o

Complement of any Angle = 90^o - Angle

Complement of ∠A = 90^o - 48^o

Complement of ∠A = 42^o

Thus, the complement of ∠A is 42^o

Example 2: Find the supplemenatry angle of ∠A = 48^o.

Solution:

Given angle,

∠A = 48^o

Supplement of any Angle = 180^o - Angle

Supplement of ∠A = 180^o - 48^o

Supplement of ∠A = 132^o

Thus, the complement of ∠A is 42^o

Example 3: Find the supplementary angle of ∠A = 98^o.

Solution:

Given angle,

∠A = 98^o

Supplement of any Angle = 180^o - Angle

Complement of ∠A = 180^o - 98^o

Complement of ∠A = 82^o

Thus, the complement of ∠A is 82^o

Example 4: Classify the angles into different categories,

1. ∠A = 12^o
2. ∠B = 172^o
3. ∠C = 232^o
4. ∠D = 180^o

Solution:

(1) ∠A = 12^o

As the measure of ∠A is less than 90^o, thus it is an acute angle.

∠B = 172^o

As the measure of ∠B is greater than 90^o, thus it is an obtuse angle.

∠C = 232^o

As the measure of ∠C is greater than 180^o, thus it is a reflex angle.

∠D = 180^o

As the measure of ∠D is equal to 180^o, thus it is a straight angle.

Practice Problems on Angles

1. Find the Supplementary Angle of ∠A = 82°

2. Find the Supplement of Angle ∠A = 108°

3. Find the Complementary Angle of ∠A = 45°

4. Find the Complement of Angle ∠A = 60°

5. Calculate the complement of a 40° angle.

6. Find the supplement of a 110° angle.

7. What type of angle is formed by the hands of a clock at 3:00?

8. Determine the measure of the angle formed by the hands of a clock at 8:00.

9. If two angles are complementary and one measures 35°, find the measure of the other angle.

10. What is the measure of an angle that is supplementary to 130°?

11. Identify the type of angle formed when two adjacent angles are each 45°.

Conclusion

Understanding angles is fundamental to the study of geometry and trigonometry, as they form the basis for numerous mathematical concepts and applications. Whether we are measuring angles in degrees or radians, exploring their properties, or applying them in real-world scenarios, angles play a vital role in various fields, including engineering, physics, and architecture. By mastering the different types of angles—acute, obtuse, right, straight, and reflex—we equip ourselves with the tools to analyze shapes, solve various problems in academics as well as in real life .