Adjacent Angles & Vertical Angles

The measure of rotation of a ray, when it is rotated about its end point is known as the angle formed by the ray between its initial and final position. At times, in geometry, the pair of angles are used. There are various kinds of pair of angles, like – supplementary angles, complementary angles, linear pair of angles, opposite angles, adjacent angles etc.

Adjacent Angles

Consider a wall clock , The minute hand and second hand of clock form one angle represented as ∠AOC and the hour hand forms another angle with the second hand represented as∠COB. Both these pair of angles i.e.∠AOC and ∠COB lies next to each other and are known as adjacent angles.

Adjacent Angles

∠AOC and ∠COB have a common vertex, a common arm and the uncommon arms lie on either side of the common arms. Such angles are known as adjacent angles.

Important:

Two angles are adjacent-angles, if

a) Share a common vertex

b) Share a common rays

c) Other rays of the two angles lie on opposite sides of the common ray

Adjacent Angles Example

Angles Not Adjacent

Adjacent Supplementary Angles

What is the sum of adjacent angles? The adjacent angles will have the common side and the common vertex. Two angles are said to be supplementary angles if the sum of both the angles is 180 degrees. If the two supplementary angles are adjacent to each other then they are called linear pair.

Sum of two adjacent supplementary angles = 180o.

Here are some examples of Adjacent angles:

Adjacent Angles Examples

Examples of Adjacent Angles

Linear Pair

Pair of adjacent angles whose measures add up to form a straight angle is known as a linear pair. The angles in a linear pair are supplementary.

Consider the following figure in which a ray \(\overrightarrow{OP}\) stand on the line segment \(\overline{AB}\)   as shown:

Linear Pair

The angles ∠POB and ∠POA are formed at O. ∠POB and ∠POA are adjacent angles and they are supplementary i.e. ∠POB + ∠POA = ∠AOB = 180°

∠POB and ∠POA are adjacent to each other and when the sum of adjacent angles is 180° then such angles form a linear pair of angles.

Vertically Opposite Angles

When a pair of lines intersects, as shown in the fig. below, four angles are formed. ∠AOD and ∠COB are vertically opposite to each other and ∠AOC and ∠BOD are vertically opposite to each other. These angles are also known as vertical angles or opposite angles.

Vertically Opposite Angles

Thus, when two lines intersect, two pair of vertically opposite angles are formed i.e. ∠AOD, ∠COB and ∠AOC, ∠BOD.

According to vertical angle theorem, in a pair of intersecting lines the vertically opposite angles are equal.

Read More:

Angles

Adjacent Angles Vertical

Vertical Angles

Complementary Angles Supplementary Angles

Examples: 

Example 1: Find the value of x.

Example on Adjacent Angles

If m∠AOB = 110° , m∠AOC = x and m∠ COB = 70°

Solution: 

From figure: 

m∠AOB = 110° 

m∠AOC = x

m∠ COB = 70° 

Now, from figure: ∠AOB = ∠AOC + ∠COB

m∠AOB = m∠AOC + m∠COB

110° = x + 70°

x = 110° – 70°

x = 30°

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Practise This Question

Three lines can never intersect at a single point.

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