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# Arm Of An Angle|Definition & Meaning

## Definition

The **arms of an angle** can be defined as **two lines** that join each other at a **common intersection** to form an **angle**. The **common intersection** is known as a **vertex**. One of the arms is usually stationary while the other one moves around to form the **angle**.

The **two arms of the angle** define the **degree of rotation** of the **angle**. One of the **arms** remains at a **fixed point** at the axis and does not move, it is known as the **stationary arm**. The second arm is free to move and rotates around the **stationary arm** around a **fixed axis**. The **vertex** is the point where both arms meet to form the **angle**.

The** stationary arm** usually remains at the x-axis. If both arms are on this axis, then the angle, by convention, is considered **zero**. From this understanding, there can be two types of movements that the stationary arm can make. It can either **revolve** in a **clockwise direction** or an **anti-clockwise direction**.

By convention, the **anticlockwise or counter-clockwise movement** is taken as a **positive movement,** whereas the **clockwise movement** is taken as a **negative movement**.

## Counter-clockwise and Clockwise Movement of the Arms

As mentioned earlier, the rotating arm can move in two directions:

- Clockwise Rotation
- Anti-clockwise or Counter-clockwise Rotation

Some conventions need to be followed to define the difference between the arm moving in either** direction**. One convention can be standardized for understanding the concept of **positive and negative angles**.

By convention, when the** stationary arm** is on the **x-axis** and the movement of the **rotating arm** is in the **clockwise direction**, the rotation is considered to be the **negative rotation** and the angle thus formed by the vertex of these arms is also taken as **negative**.

By convention, when the **stationary arm** is on the x-axis and the movement of the **rotating arm** is in the** anti-clockwise direction**, the **rotation** is considered to be the **positive rotation** and the **angle** thus formed by the **vertex** of these arms is also taken as** positive**.

## A Deeper Explanation of the Arms of an Angle

There are three basic components of an angle that need to be understood:

- Stationary Arm
- Rotating Arm
- Vertex

The **stationary arm** remains at the** x-axis**. This is the arm of reference. We can compare the rotating arm to this arm to define the difference in their position.

The **rotating arm** is the arm that is responsible for determining the **angle** that is formed between it and the **stationary arm**. It can move freely on either side of the **stationary arm**, either moving **clockwise or counter-clockwise**.

The **vertex** is the meeting or joining common point of the **stationary and rotating arms**. It defines the **angle**. It can either produce a** negative** or **positive angle** depending upon the rotation of the **rotating arm** around **the stationary arm**.

## The System of Quadrants

The** arms** lie in the 4 **Quadrants System**. If the **rotating arm** moved in either direction starting from the starting position x=0, it would cover a total of **360°**, thus making a complete rotation after reaching back to zero from either side (One can be taken as a reference).

If we move with the convention that **anti-clockwise** **rotation** is **positive**, the **angle** in the **first quadrant** will be from **0° to +90°**. It will be a **positive movement** and the coordinates of the **rotating arm** would be (x,y).

If we move in the **anti-clockwise** position further, the **angle** in the **second quadrant** will be from **0° to +180°**. It will still be a **positive movement** by convention and the coordinates of the **rotating arm** would be (-x,y).

If we move in the **anti-clockwise** position further, the angle in the **third quadrant** will be from **0° to +270°**. It will still be a **positive movement** by convention and the coordinates of the** rotating arm** would be (-x,-y).

If we move in the **anti-clockwise** position even further to complete a rotation, the **angle** in the** fourth quadran**t will be from **0° to +360°**. It will still be a **positive movement** by convention and the coordinates of the** rotating arm** would be (x,-y).

The angles would be negative with this convention if the stationary arm moves in the clockwise direction. it would be a -360 for a complete clockwise rotation.

## Illustrations of Arms of an Angle With Some Unique Angles

As we have discussed that the rotating arm of the **angle** can be revolved around the **quadrant system** to get a **complete rotation** and the complete is divided into **360 Degrees** (From **0° to 360°**). There is specific and unique nomenclature for the **angles** formed along the **quadrant system**.

**Acute Angle**

When the **rotating arm** lies in the **first quadrant**, the angle can range from** 0° to 90°**. Any angle between **0° to 90°** is known as the **acute angle**. It is represented as:

**Acute angle = 90° > α > 0°**

**Right Angle**

When the **rotating arm** lies on the edge of the **first and second quadrants**, the **angle** can range from **0° to 90°**. Any angle that is exactly **90°** is known as the **right** **angle**. It is represented as:

**Right angle = α = 90°**

**Figure 8** represents a right angle.

**Obtuse Angle**

When the **rotating arm** lies in the **second quadrant**, the** angle** can range from **90° to 180°**. Any angle between **90° to 180°** is known as the **obtuse angle**. It is represented as:

**Obtuse angle = 180° > α > 90°**

**Straight Angle**

When the rotating arm lies on the edge of the **second and third quadrants**, the angle can range from **90° to 180°**. Any angle that is exactly **180°** is known as a **straight angle**. It is represented as:

**Straight angle = α = 180°**

**Figure 9** represents a straight angle.

**Reflex Angle**

When the **rotating arm** lies in the third quadrant, the** angle** can range from **180° to 270°**. Any angle between **180° to 270°** is known as the** obtuse angle**. It is represented as:

**Reflex angle = 270° > α > 180°**

## Understanding Arms of an Angle With Examples

Consider the following angles:

- 87°
- 99°
- 267°
- 360°
- 180°
- 90°

Kindly please identify each of the following angles based on their uniqueness.

### Solution

1) 87°

As we can see that this **angle** lies in the** first quadrant** and follows the relation:** 90° > α > 0°**, we can easily identify it as an **acute angle**.

2) 99°

As we can see that this **angle** lies in the **second quadrant** and follows the relation: **180° > α > 90°**, we can easily identify it as an **obtuse angle**.

3) 267°

As we can see that this **angle** lies in the **third quadrant** and follows the relation: **270° > α > 180°**, we can easily identify it as a **reflex angle**.

4) 360°

As we can see that this **angle** lies in the **fourth quadrant** and has completed **a full rotation**, we can easily identify it as **a complete angle or a complete revolution**.

5) 180°

As we can see that this **angle** lies on the edge of the **second and third quadrants** and has completed a **half rotation**, we can easily identify it as** a straight angle or a half revolution**.

6) 90°

As we can see that this **angle** lies on the edge of the **first and second quadrants** and has completed a **quarter of a rotation**, we can easily identify it as a **right angle**.

*All images used in this article were made with GeoGebra.*