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,Â adjacent angles, linear pair of angles, opposite angles, etc. In this article, we are going to discuss the definition of adjacent angles and vertical angles in detail.

## 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 âˆ AOC and the hour hand forms another angle with the second hand represented asâˆ COB. Both these pair of angles i.e.âˆ AOC and âˆ COB lie next to each other and are known as adjacent angles.

âˆ AOC and âˆ 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

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## 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 = 180^{o}.

Here are some **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 rayÂ \(\overrightarrow{OP}\) stand on the line segment \(\overline{AB}\)Â as shown:

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

âˆ POB and âˆ POA are adjacent to each other and when the sum of adjacent angles is 180Â° 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. âˆ AOD and âˆ COB are vertically opposite to each other and âˆ AOC and âˆ 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. âˆ AOD, âˆ COB and âˆ AOC, âˆ 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 **mâˆ AOB = 110Â°Â , mâˆ AOC = x and mâˆ COB = 70Â°

**Solution:Â **

From figure:Â

mâˆ AOB = 110Â°Â

mâˆ AOC = x

mâˆ COB = 70Â°Â

Now, from figure: âˆ AOB = âˆ AOC + âˆ COB

mâˆ AOB = mâˆ AOC + mâˆ COB

110Â°Â = x + 70Â°

x = 110Â°Â – 70Â°

x = 30Â°

**Example 2:**

In the given figure, isÂ âˆ 1 adjacent toÂ âˆ 2. Give justification.

**Solution:**

In the given figure,Â âˆ 1 does not share the vertex ofÂ âˆ 2.

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

âˆ 1 is not adjacent toÂ âˆ 2.

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