Adjacent Angles & Vertical Angles

The measure of rotation of a ray, when it is rotated about its endpoint 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, adjacent angles, linear pair of angles, opposite angles, etc.

Adjacent Angles Definition

The two angles are said to be adjacent angles when they share the common vertex and side. The endpoints of the ray form the side of an angle is called the vertex of a angle. Adjacent angles can be a complementary angle or supplementary angle when they share the common vertex and side.

Adjacent Angle Example

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 lie next to each other and are known as 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.

Properties of Adjacent Angles

Some of the important properties of the adjacent angles are as follows:

Two angles are adjacent-angles, such that

  • They share the common vertex
  • They share the common arm
  • Angles do not overlap
  • It does not have a common interior-point
  • It can be complementary or supplementary angles when they share the common vertex.
  • There should be a non-common arm on both the sides of the common arm

        

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:

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:

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 intersect, 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.

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.

Adjacent Angle problem

Example 1: Find the value of x.

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°

Example 2:

In the given figure, is ∠1 adjacent to ∠2. Give justification.

Solution:

In the given figure, ∠1 does not share the vertex of ∠2.

As it does not obey the important property of adjacent angles,

∠1 is not adjacent to ∠2.

Frequently Asked Questions

What are Adjacent Angles?

Adjacent angles can be defined as two angles that have a common vertex and a common side. Two adjacent angles can be either complementary or supplementary based on their sum value.

What are Vertical Angles?

Vertical angles are defined as the angles opposite to each other when two lines cross (i.e. intersect). It is also known as vertically opposite angles. It should be noted that two vertical angles are always equal.

Can Vertical Angles be Adjacent?

No, vertical angles can never be adjacent. Adjacent angles are the ones next to each other while vertical angles are opposite from each other.