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What is an Angle in Geometry – Definition, Properties & Measurement

what is an angle in geometry

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Once you have learned about points, lines, line segments, and rays, the next thing to know is what happens when two lines meet at a point. The figure that is formed when two lines meet or intersect is an angle. Angle is one of the basic concepts of geometry. It is a part of every geometric figure be it triangles, quadrilaterals, or polygons. Even in our daily lives many objects in the shape of angles such as cloth hangers, arrowheads, scissors, partly opened doors, the edge of a table, the edge of a ruler, etc.

what is an angle in geometry

The word angle came from the Latin word “Angulus” meaning corner. Let’s understand what is an angle in geometry and its properties and how it is measured.

What is an Angle in Geometry – Angle Definition

An angle is a combination of two rays (half-lines) with a common endpoint. The latter is known as the vertex of the angle and the rays as the sides, sometimes as the legs, and sometimes as the arms of the angle. The angle is represented by the symbol $’\angle’$. 

Parts of an Angle

An angle has two main parts.

  • Vertex: The corner point of an angle is known as the Vertex. It is the point where two rays meet.
  • Arms: The two sides of the angle, joined at a common endpoint is known as the arm. The arms are of two types.
    • Initial Side: It is also known as the reference line. All the measurements are done taking this line as the reference.
    • Terminal Side: It is the side up to which the angle measurement is done.
what is an angle in geometry
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How to Label the Angles?

There are two different ways to label the angles. They are:

Method 1: The first method of labeling an angle is by giving it a name. Generally, the angle is named using the lower case letter like “a”, “x”, etc or by using the Greek letters alpha ($\alpha$), beta ($\beta$), theta ($\theta$), etc.

Method 2: In the second method, the angle is named by using the points lying at the vertex and the two arms(or the legs). The middle letter should be the vertex, which is an actual angle. 

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For example, $angle \text{ABC}$ can also be written as $\angle \text{B}$. Similarly, $\angle \text{PQR}$ can also be written as $\angle \text{Q}$.

Measuring an Angle

By measure of the angle we mean the amount of rotation about the point of intersection of two planes (or lines) that is required to bring one in correspondence with the other. There are three types of measurements in angles. An angle is measured in terms of degree $\left(^{\circ} \right)$, radians, or gradians.

While measuring an angle, is done either in a clockwise or counterclockwise(anticlockwise) direction. Based on this the measure of an angle can be negative or positive.

  • Positive Angle: An angle measured in an anticlockwise direction is called a positive angle.
  • Negative Angle: An angle measured in a clockwise direction is called a negative angle.
what is an angle in geometry

Among the above-mentioned three types of measurements, the two most commonly used measurements are

  • Degree Measurement 
  • Radian Measurement

Degree of an Angle

The degree of an angle is represented by $^{\circ}$ (read as a degree). It is a base 60 (Sexagesimal) number system. In this system, a circle is divided into $360$ equal parts, each part representing $1^{\circ}$. Therefore, an angle is said to be equal to $1^{\circ}$ if the rotation from the initial to the terminal side is equal to $\frac{1}{360}$ of the full rotation.

A degree is further divided into minutes and seconds. $1^{′}$ (1 minute) is defined as one-sixtieth of a degree and $1^{”}$ (1 second) is defined as one-sixtieth of a minute. Thus, $1^{\circ}= 60^{′} = 3600^{”}$.

what is an angle in geometry

Radian of an Angle

Radian is the SI unit of angle. Radian is mostly used in Calculus. It is denoted by ‘rad’. The length of the arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends. In a complete circle, there are $2 \pi$ radians.

Therefore, $360^{\circ} = 2 \pi \text{ radian}$.

$=> 1 \text{ radian} = \frac{180^{\circ}}{\pi}$.

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Types of Coordinate Systems

Regions in an Angle

To enable comparison and addition, an angle is associated with two regions into which the two sides of the angle split the plane. One of these is termed the interior and the other is the exterior of the angle. The region lying between the two arms (initial and terminal) of an angle is called the interior of an angle and the region lying outside the two arms is called the exterior region.

what is an angle in geometry

In order to compare the angles they should be placed so their interiors intersect while some two sides and the vertices coincide. The angle whose other side is located in the interior of the other angle is declared (and naturally so) the smaller of the two.

what is an angle in geometry

In addition, we overlap one side of one angle with a side of the other so as to insure that their interiors do not intersect. The two free sides (one from each of the addends) form an angle which is declared the sum of the two.

what is an angle in geometry

Real-life Application of Angles

The following are some of the real-life applications of angles.

  • Engineers construct buildings, bridges, houses, monuments, etc., using angle measurement.
  • Athletes use this concept in sports to enhance their performance. 
  • Carpenters use it to make equipment like doors, chairs, sofas, tables, etc. 
  • Artists use their measurement knowledge to sketch or create art pieces. 
  • Wall clocks use the concept of angles to show time with hour and minute hands.

Practice Problems

  1. Define the term angle.
  2. What are the different parts of an angle?
  3. What is the degree measure of the angle?
  4. What is the radian measure of the angle?
  5. How many degrees is equal to 1 radian?
  6. How many radians is equal to 1 degree?
  7. What is the meaning of positive and negative angle?
  8. What are the interior and exterior of an angle?

FAQs

Define angle in Geometry.

An angle is defined as the figure, in which two rays meet at a common point called the vertex. The angle is denoted using the symbol $’\angle’$.

Mention the properties of angles.

The two main important properties of angles are:
a) The sum of all the angles on one side of a straight line always measures 180 degrees.
b) The sum of all the angles around a point always measures 360 degrees.

What is meant by the interior and exterior of an angle?

The region lying between the two arms (initial and terminal) of an angle is called the interior of an angle and the region lying outside the two arms is called the exterior region.

Conclusion

An angle is a combination of two rays (half-lines) with a common endpoint. It consists of two parts – vertex and arms. The arms are of two types – initial and terminal. The two most commonly used units of measuring angles are degree and radian.

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