Contents

# Adjacent|Definition & Meaning

## Definition

The word “**adjacent**” is defined as lying **next to** or **close to** one another. It sometimes may not imply **physical contact** but certainly suggests that there is nothing in between the two entities of the same kind.

**Figure 1** shows two balls lying **adjacent** to each other.

Adjacent implies that the objects are placed in **close vicinity**. It does not suggest an **overlap** between the two objects meaning both objects have their separate physical space but are just attached through some means.

## Explanation of Adjacent

**Adjacent** means **neighboring** to or **adjoining** to something. It is used as an adjective specifying the **nearness** of one entity to another.

For **example**, it can be used in a sentence as “The coffee shop is adjacent to the beauty parlor”.

In mathematical** geometry**, “**adjacent**” is used for the **sides** or **angles** which are joined by a common point, corner, or an end called a vertex. **Vertices** can also be adjacent to each other if they are adjoined by a line.

## Adjacent Angles and Linear Pairs

**Adjacent angles** are important while dealing with linear pairs. Two adjacent angles are **linear pairs** if the sum of the adjacent angles equals **180°**.

The two angles are** supplementary** with rays in the **opposite** **direction** not pointing to each other.

## Non-Adjacency

Two angles are** not adjacent** if they are not opposed to each other from the common **boundary line** or side of the angle.

If the angles have a **common side** and** vertex** but are not facing opposite to each other but instead are **overlapping**, they are not considered adjacent angles.

## Adjacent Sides, Angles, and Vertices

The word “**adjacent**” can be used for sides, angles, and vertices while dealing with **geometrical shapes**. The **definitions** for adjacent sides, adjacent angles, and adjacent vertices are as follows.

### Adjacent Sides

**Adjacent sides** are defined as the **sides** which are connected by a point known as the **vertex**. One end of both sides is **joined** by a common **corner** which makes them adjacent to each other.

Adjacent sides can be found in geometrical **shapes** such as triangles, rectangles, parallelograms, or trapezoids involving **line segments**.

### Adjacent Angles

An **angle** contains a vertex and two sides. Two angles having a **common side** and** vertex** are known as adjacent angles. In addition to this, both angles should not have **common interior points**.

**Adjacency** does not allow overlapping. The two adjacent angles cannot be **overlapped** over one another. This is the concept of the two angles not having the same **interior points**.

All **three conditions** should be met for the angles to be adjacent to each other.

### Adjacent Vertices

A **vertex** is a point where two rays or two lines meet. **Vertices** can also be **adjacent** to each other. Two vertices are adjacent if they are **joined directly** by a line segment

In a square or rectangle, the **corners** are known as vertices. The pair of **vertices** joined by the sides of the square or rectangle are **adjacent** to each other.

## Demonstration of Adjacent Sides, Angles, and Vertices

To understand the meaning of “**adjacent**”, consider the following illustrations for adjacent **sides**, adjacent **angles**, and adjacent **vertices**.

Consider **figure 2** for the **adjacent sides**. The sides **AB** and **BD** are adjacent to each other as they are connected by a common vertex **B**. Similarly, the sides **AC** and **CD**, and **BD** and **CD** are also adjacent.

But the sides **AB** and **CD** are not adjacent to each other as they do not share a common point.

Consider **figure 3** for **adjacent angles**. The figure shows three angles; angle **PSQ** = **a**, angle **QSR** = **b**, and angle **PSR** = **a + b**. The angles **PSQ** and **QSR** are adjacent to each other as they share a common side **QS**.

They also share a **common** **vertex S** and are not overlapping over one another thus one angle does not have interior points in the other angle.

The **angles** **PSR** and **QSR** have a common **side** that is **SR** and do have a common **vertex** **S** but are non-adjacent angles. This is because the angle **QSR** has interior points in angle **PSR** which shows an overlap between them.

Adjacent angles are opposite to each other. An **angle bisector** can make one angle into two adjacent angles.

Consider **figure 4** for **adjacent vertices**. **L**, **M**, **N**, and **O** are vertices in the rectangle **LMONL**. Here **f**, **g**, **h**, and** i** are the edges or sides of the rectangle. The diagonal **j** can only be considered as an edge.

The **vertices** **L** and **M** are adjacent to each other as they are directly joined by a **side** **g**. Similarly, the vertices **N** and **M** are also **adjacent** to each other as they contain an edge** j** joining them.

Adjacent vertices are also known as **end-vertices**. The vertices **L** and** O** are **non-adjacent** vertices as they are not connected directly by an edge.

## Examples

### Example 1 – Identifying Adjacent Sides

**Figure 5** given below shows a triangle **EFGE**. Highlight the **adjacent sides** in the given triangle and explain what makes them adjacent.

### Solution

The triangle contains the sides **EF**, **FG**, and **GE**. The sides **EF** and **FG**, **FG** and **GE**, and **GE** and **EF** are **adjacent** to each other. This is because they all contain a common corner or vertex **F**, **G**, and **E** respectively.

### Example 2 – Identifying Adjacent Angles

The following **figure 6** shows three angles; angle **XWY** = **q**, angle **YWZ** = **p**, and angle **XWZ **= **q + p**. Show which angles are **adjacent** and which are **non-adjacent** to each other.

### Solution

The angles **XWY** and **YWZ** are adjacent to each other as they share a common **vertex** **W** and a common **side** **YW**. They also are connected in opposite directions to each other.

Whereas, angles **XWY** and **XWZ**, and angles **YWZ** and **XWZ** are non-adjacent angles as they are overlapping with each other.

### Example 3 – Identifying Adjacent Vertices

**Figure 7** shows a square **ABCDA**. Point out the **adjacent vertices** and one non-adjacent vertex.

### Solution

The vertex **A** and **B**, **B** and **C**, **C** and **D**, and **D** and **A** are **adjacent **vertices with **b**, **c**, **d**, and **a** as the edges or sides joining the vertices.

*All the geometrical figures are created using GeoGebra.*