Definition of Angles: Introduction, Types and Examples

When it comes to stating the definition of angles, an angle can be defined as the figure formed by two rays meeting at a common endpoint. In simple words, an angle is a shape, formed by two lines or rays diverging from a common point (the vertex). An angle is represented by the symbol ∠. The concept of angle is a very important concept for all of Greek geometry.

Coming to sports, athletes use angles to enhance their performance, such as steering the car, throwing a baseball, shooting the ball into the basket, kicking the soccer ball, etc. In this article, we shall learn more about angles.

Definition of Angles

When two rays or lines intersect at a point, the measure of the region (opening) between these two rays or lines is called an “Angle”. 

It is denoted by using the symbol \(\angle .\) Angles are usually measured in radians \({\pi ^c}\) and degrees \(\left( {^{\rm{o}}} \right) \cdot \left[ {{\pi ^c} = {{180}^{\rm{o}}}} \right]\) 

The word “angle” has originated from the Latin word “angulus”, which means “corner”. The rays or lines are called the arms or sides of an angle, and the common endpoint is called the vertex.

Definition of Parts of Angles

In a two-dimensional geometry, an angle is formed when two rays are started from the same starting point. There are different parts of the angle, which are listed below:

1. Vertex
2. Arms
3. Initial side
4. Terminal side

Arms

The rays or lines of the angles are known as the sides or arms of the angle. Observe the below figure, in which rays \(OA, OB\) are the sides of the angle.

Vertex

The vertex of the angle is the common endpoint, where two rays of the angles end. It is also called as “node” of the angle. Observe the below figure, in which two rays are meeting at the vertex.

It is the base or reference side of the angle. All the measurements of the angle are made by taking this line as the base for reference.

Terminal Side

It is the side or ray where we can measure the angle.

Types of Angles

There are different types of angles based on their measurement, which are mentioned below:

Zero Angle

An angle that is equal to \({{0^{\rm{o}}}}\) is called a zero angle. In the diagram below, \(PQ\) and \(PR\) are two rays originating from the same point and are in the same direction. Hence, it represents zero angles.

Acute Angle

An angle whose measure is between \({{0^{\rm{o}}}}\) and \({{90^{\rm{o}}}}\) is called an acute angle.

Right Angle

If the two sides of an angle are perpendicular to each other, then the angle between them measures exactly \({{90^{\rm{o}}}}\). An angle that measures exactly \({{90^{\rm{o}}}}\) is called a right angle.

Obtuse Angle

An angle whose measure is between \({{90^{\rm{o}}}}\) and \({{180^{\rm{o}}}}\) is called an obtuse angle.

Straight Angle

If the arms of the angles are stretched in the opposite direction, they form a straight angle. Thus, one straight angle equals the sum of two right angles.

Reflex Angle

An angle that is greater than \({{{180}^{\rm{o}}},}\) but less than \({{360^{\rm{o}}}}\) is known as a reflex angle.

Complete Angle

An angle whose measure is exactly \({{360^{\rm{o}}}}\) is known as a complete angle. It is equivalent to four right angles. It is also called a full angle.

Complementary Angles

Two angles are said to be complementary angles if the sum of the two angles is \({{{90}^{\rm{o}}}.}\) For each of the pair of complementary angles, one angle is said to be the complement of another angle.As shown in the diagram below, the angle \({{30^{\rm{o}}}}\) is the complement of the angle \({{60^{\rm{o}}}}\) or vice versa because their sum is \({{90^{\rm{o}}}}\)

Supplementary Angles

Two angles whose sum is \({{180^{\rm{o}}}}\) are known as supplementary angles. If two angles are supplementary, each angle is called the supplement of the other.As shown in the diagram below, the angle \({{60^{\rm{o}}}}\) is the supplement of \({{120^{\rm{o}}}}\) angle or vice versa because their sum is \({{{180}^{\rm{o}}}.}\)

Linear Pair

The straight lines form a linear pair of angles. Two adjacent angles on the straight line are said to be in linear pair if their sum is \({{{180}^{\rm{o}}}.}\)

A straight angle measures \({{{180}^{\rm{o}}}.}\) So, a linear pair of angles must be added up to \({{{180}^{\rm{o}}}.}\)In the below figure \(\angle 1\) and \(\angle 2\) are called the linear pair of angles.

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 have the same vertex and the same common side.

Angles at a Point

Angles about a point add up to \({{360^{\rm{o}}}}\) (which is a complete circle).

If a point is on a straight line, then the angles on each side of the line add up to \({{{180}^{\rm{o}}}.}\)

Angle Bisector

An angle bisector is a ray that divides a given angle into two equal angles.

Angles Made by a Transversal

When a transversal intersects two parallel lines, there is a total of \(8\) different angles formed.

Different types of angles formed by a transversal are given below:

Corresponding Angles

Corresponding angles are formed at the same corners of the transversal. The pair of corresponding angles formed by a transversal with parallel lines are equal In nature.The pair of corresponding angles given in the below diagram are \(\angle 1,\) and \(\angle 5,\angle 2\) and \(\angle 6,\angle 3\) and \(\angle 7,\angle 4\) and \(\angle 8.\)

Alternate Interior Angles

The angles, which lies an interior portion of the transversal with parallel lines in opposite positions, are known as alternate interior angles.

In the given figure, \(\angle 4\) and \(\angle 5\) are the pair of alternate interior angles. Similarly, \(\angle 3\) and \(\angle 6\) also the pair of alternate interior angles, which are equal in nature.

Alternate Exterior Angles

The angles, which lies an exterior portion of the transversal with parallel lines in opposite positions, are known as alternate exterior angles.

In the given figure, \(\angle 1\) and \(\angle 8\) are the pair of alternate exterior angles. Similarly, \(\angle 2\) and \(\angle 7\) also the pair of alternate exterior angles, which are equal in nature.

Vertical Angles

When two lines intersect each other, then the angles formed at the opposite side at the same vertex are called vertically opposite angles.

The angles 1 and \(\angle 4,\angle 2\) and \(\angle 3,\angle 5\) and \(\angle 8,\angle 6\) and \(\angle 7\) are Vertically Opposite angles as shown in the diagram below.

Co-Interior Angles

The angles that lie on the same side of the transversal line and are positioned at the inner corners like angles \(\angle 3\) and \(\angle 5,\) and angles \(\angle 4\) and \(\angle 6\) are co-interior angles. Their sum is \({{{180}^{\rm{o}}}.}\)

Co-Exterior Angles

The angles that lie on the same side of the transversal line and are positioned at the exterior of the parallel lines like the angles \(\angle 1\) and \(\angle 7,\)  and the angles \(\angle 2\) and \(\angle 8\) are co-exterior angles. Their sum is \({{{180}^{\rm{o}}}.}\)

Solved Examples – Definition of Angles

Q.1. How many right angles make the angle \({{{360}^{\rm{o}}}}\)?
Ans: We know, \({{{90}^{\rm{o}}}}\) is equivalent to \(1\) right angle.
Hence, \({{{360}^{\rm{o}}}}\) is equivalent to \(\frac{{{{360}^{\rm{o}}}}}{{{{90}^{\rm{o}}}}} = 4\) right angles.

Q.2. In the adjoining figure, what value of \(x\) will make \(AOB\) a straight line?

Ans: \(AOB\) will be a straight line if \(\angle AOC + \angle BOC = {180^{\rm{o}}}.\)
Therefore, \({(3x + 5)^{\rm{o}}} + {(2x – 25)^{\rm{o}}} = {180^{\rm{o}}}\)
\( \Rightarrow {(5x – 20)^{\rm{o}}} = {180^{\rm{o}}}\)
\( \Rightarrow 5x = {180^{\rm{o}}} + {20^{\rm{o}}} = {200^{\rm{o}}}\)
\( \Rightarrow x = \frac{{{{200}^{\rm{o}}}}}{5} = {40^{\rm{o}}}\)
Hence, \(x = {40^{\rm{o}}}\) will make \(AOB\) a straight line.

Q.3. Find the angle which is its own complement.
Ans: Let the required angle be \(x.\) The sum of two complementary angles is \({90^{\rm{o}}}.\)
Then, \(x + x = {90^{\rm{o}}}\)
\( \Rightarrow 2x = {90^{\rm{o}}}\)
\( \Rightarrow x = \frac{{90}}{2} = {45^{\rm{o}}}\)
Hence, the required angle measures \({45^{\rm{o}}}.\)

Q.4. What is the instrument used for measuring an angle?
Ans: The instrument used to measure the size of an angle is a protractor. The standard size of a protractor is \({180^{\rm{o}}}.\) There are two sets of numbers on a protractor.
1. One in a clockwise direction
2. Another in an anti-clockwise direction

Q.5. In the figure given below, \(BD\) is the bisector of \(\angle ABC\) and \(BE\) bisects \(\angle ABD.\) Find the measure of \(\angle DBE\) given that \(\angle ABC = {80^{\rm{o}}}.\)

Ans: It is given that \(\angle ABC = {80^{\rm{o}}}.\) \(BD\) is an angle bisector bisecting \(\angle ABC\) in two equal parts.
\(\angle ABD = \frac{1}{2} \times \angle ABC = \frac{1}{2} \times {80^{\rm{o}}} = {40^{\rm{o}}}\)
Now, \(BE\) is a bisector and bisects \(\angle ABD\) in two equal parts
\(\angle DBE = \frac{1}{2} \times \angle ABD = \frac{1}{2} \times {40^{\rm{o}}} = {20^{\rm{o}}}\)
\(\therefore \angle DBE = {20^{\rm{o}}}\)

Summary

In this article, we have discussed the definition of angles, parts, and units of measurement of angles. Here, we have also discussed types of angles and angles formed by a transversal, such as corresponding angles, interior and exterior angles etc. These will help students to understand geometry problems better.

Frequently Asked Questions (FAQ) – Definition of Angles

Q.1. What are the adjacent angles?
Ans: The two angles which have a common vertex and a common side but they do not overlap are known as adjacent angles.

Q.2. Can an angle have more than one angle bisector?
Ans: No, an angle can have only one angle bisector. For example, if we bisect \({60^{\rm{o}}}\) angle, we will get \({30^{\rm{o}}}\) as a result. This means \({60^{\rm{o}}}\) angle is divided into two equal angles \({30^{\rm{o}}}\)(each).

Q.3. What are 6 types of angles?
Ans: There are many types of angles. Among them, \(6\) types of angles are acute angle, obtuse angle, right angle, straight angle, reflex angle and complete angle.

Q.4. What are angles?
Ans: The angle is the region formed between two rays, which are starting from the same point, which is known as the vertex of the angle.

Q.5. What are 5 types of angles?
Ans: There are many types of angles. Among them, \(5\) types of angles are acute angle, obtuse angle, right angle, straight angle, and reflex angle.

Q.6. What is the tool to measure the angle?
Ans: The easiest and the common way to measure the angles is to use a protractor.

We hope you find this detailed article on angles and their definition helpful. If you have any doubts or queries regarding this topic, feel to ask us in the comment section.