# Angles

An angle is a shape that is formed when two rays have the same endpoint. The rays forming an angle are called the arms of the angle and the point of intersection of the two arms is called the vertex of the angle. Angles are denoted by the symbol ‘∠’.

In the image given below, O is the vertex; ray OA and ray OB are the arms of the angle.

Angle is denoted by a symbol ‘∠’. In the image above, ∠AOB is the angle formed by rays OA and OB. An angle is measured by the amount of rotation, or turning, from 1 ray to another. Protractor is the device used to measure the angle.

## Types of Angles

Angles are classified based on their measurement. The table below describes the classification.

Angle Name
Description
Example
Acute angle
Angle whose measure is greater than 0° and less than 90° is called an acute angle. Obtuse angle
Angle whose measure is greater than 90° and less than 180° is called an obtuse angle. Right angle
Angle whose measure is exactly 90° is called a right angle. Straight angle
Angle formed by two opposite rays and measures 180° is called a straight angle. Reflex angle
Angle whose measure is more than 180° and less than 360° is called a reflex angle. Full rotation
Angle whose measure is exactly 360° is called a full rotation. How do you name the angle?

Consider the angle shown in the image below. It is named in any of three different ways listed below:

• Using the vertex and a point on each ray where vertex should be at the center, ∠ABC or ∠CBA.
• Using only vertex, when there is only one possible angle, ∠B.
• Using a number, ∠1.
•
Remember!

You cannot name the angle just by a vertex, when multiple angles are present at that vertex. Refer to the image below. There are two angles formed at vertex O. Thus, you cannot name the angles using only O. Here, naming can be done as ∠LOM or ∠MOL, ∠NOM or ∠MON, ∠LON or ∠NOL.

Challenge yourself!

Identify the type of angle shown in each of the images given below.
Options: Acute, Obtuse, Right.

How do you measure the angle using a protractor?

To measure the angles using the protractor, follow the steps given below:

1. Place the center of the protractor on the endpoint of the angle, with the straight line along one ray.

2. Now, one ray will be lined up with zero.

3. Find the tick mark on the protractor that lines up with the second ray of the angle. For example, in the image given below, the angle measure is 95°.

## Pairs of Angles

Angles are classified into different types based on the position of pairs of angles.

In the image given below, the two angles ∠1 and ∠2 in each of the angle representation have common vertex and a common arm. Such a pair of angles is called adjacent angles.

### Vertically opposite angles

In the image given below, two lines intersect each other. The pair of angles formed vertically opposite is equal and are called vertically opposite angles.
That is, vertically opposite angles are ∠1 = ∠2 and ∠3 = ∠4.

Remember! When a transversal intersects two lines (parallel or non-parallel), the following angles are formed. Refer to the image below.

 ∠1 and ∠5 ∠2 and ∠6 ∠3 and ∠7 ∠4 and ∠8 Corresponding angles ∠3 and ∠6 ∠4 and ∠5 Alternate angles ∠3 and ∠5 ∠4 and ∠6 Interior angles ∠1 and ∠4 ∠2 and ∠3 ∠5 and ∠8 ∠6 and ∠7 Vertically opposite angles ∠1 and ∠2; ∠3 and ∠4 ∠5 and ∠6; ∠7 and ∠8 ∠1 and ∠3; ∠2 and ∠4 ∠5 and ∠7; ∠6 and ∠8 Adjacent angles

## Properties of Angles

There are certain properties of angles that describe their uniqueness.

• All right angles and straight angles are congruent respectively.
• In a triangle the sum of the angles is 180°.
• The two angles are said to be supplementary, when the sum of the measure of angles equals 180°. These angles for a linear angle.
• The two angles are said to be complementary, when the sum of the measure of angles equals 90°.
•

### Problems on Angles

Example 1: Find the complementary and supplementary angles of the following angles.
a) 32° b) 57° c) 45°

Solution:

Angle
Complementary angle
Supplementary angle
32°
58°
122°
57°
33°
147°
45°
45°
135°

Example 2: Refer to the image given below. If AC and CB are opposite rays, then find the following:

a) For x = 16, what is the value of y?

b) For x = 17.5, what is the value of y?

Solution: The sum of the angles ∠BCD and ∠ACD is supplementary.
a) x = 16 ⇒ 4x = 4 × 16 = 64°
(4x)° + (7y + 1)° = 180°
64° + 7y + 1 = 180°
7y = 180 – 65
7y = 115 ⇒ y = 16.43

b) x = 17.5 ⇒ 4x = 4 × 17.5 = 70°
(4x)° + (7y + 1)° = 180°
70° + 7y + 1 = 180
7y = 180 – 71
7y = 109 ⇒ y = 15.57

## Math Facts

• In the construction industry, to ensure the building is safe or not, angles play a very important role. To create a structure which stands upright and allows rainwater to run off the roof, architects and contractors need to calculate angles very accurately.
• Athletes use angles to enrich their performance. Carpenters use angles to design chairs, tables, and sofas.
•

## Where are angles used?

Here are some examples where we use angles in our day to day life.

• In the construction industry, to ensure the building is safe or not, angles play a very important role. To create a structure which stands upright and allows rainwater to run off the roof, architects and contractors need to calculate angles very accurately.
• Athletes use angles to enrich their performance. Carpenters use angles to design chairs, tables, and sofas.
•

## Mathemagician

Eudemus has made remarkable achievements in this archive since he has been the first major historian of mathematics. The most important of Eudemus’ three mathematical histories is the History of Geometry. Despite the fact that the work did not survive, it was available to many following writers who used it extensively. We are fortunate, then, that much of Eudemus’ knowledge of the history of Greek mathematics prior to Euclid (it had to be prior to Euclid given the dates when Eudemus was writing) has survived, despite the fact that his work has not. We have quoted from accounts based on Eudemus in many of the entries in this site. To give an example, we only know of Hippocrates’ work on the quadrature of lunes because of Eudemus’ History of Geometry.

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