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Angle and its measurements.

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Be it for geometry or trigonometry, one should understand the significance and importance of angles. Have you ever wondered what makes up an angle? Or, why do we need to study different types of angles? There are multiple reasons for which a person uses angles.

For instance, we study angles and their measurements in geometry. However, trigonometry is also a science that involves the study of triangles and the different angles involved with them. In geometry, one study is different types of shapes along with their interior angles. And to study them, one should know how to measure them. Keep reading this article to find out how to measure different angles. But first, let us get an in-depth detail about angles.

What do you understand by an angle?

In simple words, an angle is defined as a structure that is formed when two lines meet at a certain point. This common point is known as a vertex. In geometric terms, an angle is formed when two rays of line join at a specific point such that they represent an angle between them.

Even though two straight lines are needed to form an angle, the particular angle formed is common to both lines. These two lines are defined as the sides of an angle. Therefore, they share the common vertex and the angle formed between them.

What are the different parts of an angle?

As mentioned above, two main parts comprise an angle. They are –

  •  Arms of an angle
  •  Vertex

Arms of an angle

The arms of an angle refer to the pair of line segments that join each other at a certain point to produce an angle. These lines are called arms of the angle. Even if the two lines touch each other to form a straight line, a solid angle is formed between them (180° is the angle formed on a straight line).

Vertex of the angle

When the arms of an angle meet at a certain point, their point of contact is known as a vertex. This common point initiates the formation of the angle between the two arms of lines.

What are the types of angles?

To understand the measurement of an angle, one should also know the common types of angle and their properties. There are six different angles formed when two lines join each other. 

Every angle formed by two straight lines has unique features and characteristics that help identify the type of angle. Based on the specific features and degree of an angle, the types of angles are –

  •  Obtuse angle
  •  Acute angle
  •  Right angle
  •  Straight angle
  •  Complete angle
  •  Reflex angle

Explain the different types of angles

Even though all the angles are made above arms and are formed at the vertex of the two lines, they possess different features that make them categorized into six different types of angles.

  •  Obtuse angles

And the obtuse angle is formed when the arms of an angle form an angle which is more than 90°. Nevertheless, this angle is greater than 90° but less than 180°.

  •  Acute angle

This type of angle is the opposite of the obtuse angle. Meaning this angle is formed when two lines meet at a vertex and form an angle of more than 0° but less than 90°. Acute angles lie somewhere between 0° and 90°.

  •  Right angle

A right angle is formed when the two arms joining at a vertex are perpendicular to each other. Meaning they make a right angle or an exact angle of 90° between them. This type of angle lies somewhere in between an acute angle and an obtuse angle. A degree less makes this angle an acute angle, and a degree more makes this angle an obtuse angle.

  •  Straight angle

This angle is formed when the two line segments meet at a certain point; however, they create a straight line. When measured, the angle formed with two straight lines is always 180°. In other words, if there is an angle of 180° between two lines, it is safe to assume that the given angle is a straight line.

  •  Complete Angle

As the name indicates, a complete angle is an angle that covers all the degrees. These types of angles are formed when a single line takes a full rotation and comes back to its original place. Thus, making an angle of 360°.

  • Reflex angle

A reflex angle is an angle formed when the two arms of the same joint such that they make an angle that is more than the angle of the straight line but less than a full angle. 

It means a reflex angle is greater than 180° but less than 360°.

How to measure an angle?

A degree is the unit of measurement used to define an angle. A complete 360° angle is supposed to be formed by one full rotation around a center. 

A protractor is the easiest and most convenient tool to measure any angle. This particular tool is shaped like a semi-circle and is used to measure angles.

Two different scales are given with a marked degree of angles in a protractor. While one scale is smart in the clockwise direction, the other scale is marked in counterclockwise order. But because a protractor is a semi-circle, it only has a limit of measuring angles up to 180°.

Why is it important to measure angles?

It is important to measure angles because measuring the correct degree of an angle can be applied to various simple and complicated applications.

The measurement of an angle is useful in activities of construction, athletics, architecture, engineering, and various things. It is interesting to know that even scientists and astronomers have to measure angles of the celestial bodies to research their movement and path of revolution.

When two lines meet at a vertex, they form an angle, and this point of contact (vertex) is used to measure the angle. Different angles are measured in units of degrees. The measurement of angles is an important segment of geometry and trigonometry.

What is Radian?

A Radian is the unit of angular measurement equal to the angle formed at the center of a circle. It is defined as an angle that subtends from the center of a circle and intercepts at an arc equal to the radius of the circle.

One radian is equal to 57.3°.

History of radian

Although James Thompson developed the term radian around 1870, other mathematicians used this method to measure their angles way before him. A mathematician, Leonhard Euler, specifically stated that angles could be measured by the length of arc made by cutting off the unit circle.

What is the relation between real numbers and radian?

To establish a relationship between real numbers and radian, one can refer to the steps below.

  • Suppose a circle with a center O.
  • Now, locate a point P on the circumference of the circle.
  • Consider the side OP as the initial arm of an angle
  • As the initial arm of an angle is OP, the length of the arc in a circle will measure the radian of the angle.
  • The radian measure of the formed angle will subtend at the middle of the circle.
  • Now, draw a straight line APB as a tangent to the circle, with P as its point of contact.
  • Suppose this straight line is a number line that has real numbers.
  • Thus, when the line PA is roped in counterclockwise motion along the circle and line PB in a clockwise motion, all the real numbers in the number line will correspond to the concerned radian measure.

What do you understand by the measures of an angle?

In geometrical terms, the measure of an angle is defined as an angle measure that is formed by the meeting of two line segments or angle arms at a common point. This common point where an angle is formed is known as the vertex.

Apart from this, angles are measured in degrees with the symbol (°). An angle is easily measurable by the use of a protractor and by other manual practices.

What is known as the standard position of an angle?

The standard position of an angle is said to be its position when the vertex is at the point of origin, and one ray of the angle is present on the positive side of the X-axis.

The line on the X-axis is known as the initial side, and the second ray of the line is defined as the terminal side of the angle.

Summary

The above information is about measuring angles and how to measure them. By following the above information, one can get a clear idea about the definition of an angle, types of an angle, and the relationship between real numbers and radians. Any type of angle can be measured for various kinds of calculations applicable in different areas of life.

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