A figure formed by joining two different rays starting from same fixed initial point is called an angle.

Example of an angle \(\angle AOB\)

In the given figure, this figure is made up of two rays \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\). The common end point of two rays is called vertex of the angle.

So, O is vertex of angle AOB.

The rays \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) are called the arms or sides of angle AOB.

An angle is denoted by symbol \(\angle \).

Only capital letters of English alphabets are used to name an angle. Name of angles can be written using three or one alphabet.

Thus, we can write the above angle in figure as \(\angle AOB\) or \(\angle BOA\) or \(\angle O\).

We can see from the naming that vertex is always kept at the center when written using three alphabets and only vertex when written as single alphabet.

The unit of measuring an angle is degree.

The word degree originates from Latin word “gradius” which means “step”. It refers to a stage in an ascending or descending order.

The symbol used for degree is “\(\ ^0\)”. It is inserted on the right top of numeral.

for example, 90 degrees = \(90^0\)

An angle which is less than \(90^o\), is called acute angle.

Example of Acute angle

An angle which is equal to \(90^o\), is called right angle.

Example of Right angle

An angle which is greater than \(90^o\) and less than \(180^o\) is called obtuse angle.

Example of Obtuse angle

An angle which is equal to \(180^o\), is called straight angle or straight line angle.

Example of Straight angle

An angle which measure greater than \(180^o\) but less than \(360^o\) is called reflex angle.

Example of Reflex angle

An angle is said to be complete angle if two different rays coincide with initial point after making a complete revolution.

Example of Complete angle

Here, ray \(\overrightarrow{OA}\) and ray \(\overrightarrow{OB}\) coincide each other after making a complete revolution.

\(\angle AOB = 360^0\)

An angle is said to be zero angle if two different rays coincide without any revolution.

Example of Zero angle

Here, ray \(\overrightarrow{OA}\) and ray \(\overrightarrow{OB}\) coincide \(\angle AOB\)

\(\angle AOB = 0^0\)

Note

The acute and obtuse angles are known as **oblique angles**.

Angles having same measure are said to be congruent angles.

Example of Congruent angles

Two angles are said to be adjacent angles if they have common vertex, a common arm and other two arms of the angles are on the opposite sides of the common arm.

Example of Adjacent angles

In the given figure, two angles \(\angle AOB\) and \(\angle BOC\) have a common arm OB, a common vertex O and other two arms OA and OC lie on the opposite sides of common arm OB.

Two angles are said to be complementary if they form adjacent angle and sum of their measure is equal to \(90^o\)

Example of Complementary angles

\(\angle AOC + \angle BOC\)

\(= 45^o + 45^o\)

\(= 90^o\)

Two angles are said to be supplementary angles if they form adjacent angles whose sum of their angles is equal to \(180^o\)

Example of Supplementary angles

\(\angle ABO + \angle CBO\)

\(=120^o + 60^o\)

\(=180^o\)

So far from above, We have learnt the basics of angles from its definition, measurement of an angle to its various types of angles. As we have seen that angles are formed only when two lines intersect at some point. So, there are many more case where different types of angles are defined when lines intersect each other.

Such angles are formed by a transversal lines when it intersects two or more parallel or non parallel lines.

When two straight lines intersect each other, they form four angles at the point of intersection. Out of four angles the two angles which are directly opposite to each other are called vertically opposite angles. These two vertically opposite angles are always equal.

Example of Vertically Opposite angles

Here, in the above diagram, we can see \(\angle AOD\), \(\angle BOC\), \(\angle AOC\) and \(\angle BOD\) are the four angles formed at point O when two lines AB and CD intersect at point O.

The angles \(\angle AOD\) and \(\angle BOC\) are directly opposite to each other, therefore they are called vertically opposite angles and \(\angle AOD = \angle BOC\)

Similarly, the angles \(\angle AOC\) and \(\angle BOD\) are also directly opposite to each other, therefore they are also called vertically opposite angles and \(\angle AOC = \angle BOD\)

Moreover, sum of each pair of adjacent angles is alway equal to \(180^0\).

\(\angle AOC + \angle BOC = 180^0\)

\(\angle AOC + \angle AOD = 180^0\)

\(\angle BOC + \angle BOD = 180^0\)

\(\angle AOD + \angle BOD = 180^0\)

Exterior angles are formed at points on the exterior sides of the two lines where a transversal line intersects these two lines.

Let’s understand it by an example.

Example of Exterior angles

\(\angle 1\), \(\angle 2\), \(\angle 7\) and \(\angle 8\) are formed at points A and B on the exterior sides of two non parallel lines l and m respectively, when a transversal line n cuts through them. Therefore, \(\angle 1\), \(\angle 2\), \(\angle 7\) and \(\angle 8\) are called as exterior angles.

Interior angles are formed at points on the interior sides of the two lines where a transversal line intersects these two lines.

Let’s understand it by an example.

Example of Interior angles

\(\angle 3\), \(\angle 4\), \(\angle 5\) and \(\angle 6\) are formed at points A and B on the interior sides of two non parallel lines l and m respectively, when a transversal line n cuts through them. Therefore, \(\angle 3\), \(\angle 4\), \(\angle 5\) and \(\angle 6\) are called as interior angles.

Alternate angles again are formed when a transversal line cuts through two parallel or non parallel lines. These angles are a pair of angles which can exist on the interior or exterior sides of the two lines.

We can understand it more precisely with the following example.

Example of Alternate angles

Here, a transversal line n cuts thru two lines l and m and forms total of eight angles at points A and B which are:

\(\angle 1\), \(\angle 2\), \(\angle 3\), \(\angle 4\), \(\angle 5\), \(\angle 6\), \(\angle 7\) and \(\angle 8\).

If we select the pairs of angles in the following way, then these pairs are the alternate angles.

\(\angle 1\) and \(\angle 7\)

\(\angle 2\) and \(\angle 8\)

\(\angle 3\) and \(\angle 6\)

\(\angle 4\) and \(\angle 5\)

Depending upon where the pair exists on the two lines l and m we can name them alternate interior angles and alternate exterior angles.

e.g. here, the pairs of alternate angles \(\angle 1\) and \(\angle 7\), \(\angle 2\) and \(\angle 8\) lie outside the lines
i.e. exterior sides of the two lines, such type of alternate angles are called **alternate exterior angles**.

Similarly, the pairs of alternate angles \(\angle 3\) and \(\angle 6\), \(\angle 4\) and \(\angle 5\) lie inside the lines
i.e. interior sides of the two lines, such type of alternate angles are called **alternate interior angles**.

Note

The pair of alternate angles formed with parallel lines are always equal.

In the above example, lines l and m are parallel lines. So, we can see that the following alternate angles pairs are equal

\(\angle 1 = \angle 7\)

\(\angle 2 = \angle 8\)

\(\angle 3 = \angle 6\)

\(\angle 4 = \angle 5\)

A pair of angeles is called corresponding angles in which one arm of both angles is on the same side of transversal and their other arm are directed in same sense.

If lines l and m are parallel, then pair of corresponding angeles are equal.

Let’s understand it with the following example.

Example of Corresponding angles

So, from the above diagram, the following pair of angles are corresponding angles that are formed on two parallel lines l and m.

\(\angle 1\) and \(\angle 5\)

\(\angle 2\) and \(\angle 6\)

\(\angle 3\) and \(\angle 7\)

\(\angle 4\) and \(\angle 8\)

As said above, these pair of angles, also called as **corresponding angles**, are equal always.

Therefore, we can write them as:

\(\angle 1 = \angle 5\)

\(\angle 2 = \angle 6\)

\(\angle 3 = \angle 7\)

\(\angle 4 = \angle 8\)

More, in parallel lines l and m, sum of interior angles of the same side of transversal is \(180^0\) i.e. \(\angle 4\) + \(\angle 5\) = \(180^0\)

\(\angle 3\) + \(\angle 6\) = \(180^0\)

When sum of two adjacent angles is \(180^0\), they are called **linear pair of angles**.

Or, we can say when supplementary angles formed on a straight line, those angles are called a **linear pair of angles**.

Example of Linear pair of angles

In the above diagram, \(\angle ACD\) + \(\angle BCD\) = \(180^0\), therefore, they form a linear pair of angles.