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.
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:
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.
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.
∠2 and ∠6
∠3 and ∠7
∠4 and ∠8
∠4 and ∠5
∠4 and ∠6
∠2 and ∠3
∠5 and ∠8
∠6 and ∠7
∠5 and ∠6; ∠7 and ∠8
∠1 and ∠3; ∠2 and ∠4
∠5 and ∠7; ∠6 and ∠8
Properties of Angles
There are certain properties of angles that describe their uniqueness.
Problems on Angles
Example 1: Find the complementary and supplementary angles of the following angles.
a) 32° b) 57° c) 45°
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
Where are angles used?
Here are some examples where we use angles in our day to day life.
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.