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Angles

Angles

Did you know that knowing the amount of space formed from two rays meeting at a point is not only relevant to construction and surveying?

This can be a very significant phenomenon also in surgery, as surgeons need to know accurately the way they move around the body.

In this article, we are going to study angles, their types, and how they are measured.

Angles definition

An angle is defined as the figure that is formed when two rays meet at a common point.

Angles are denoted by ∠ or commonly by Greek letters. They can be either measured in degrees, radians, or gradians.

Components of an angle

An angle is defined by two fundamentals: the arms of the angle and the vertex.

  • The arms of the angle are the two rays that come together to form the angle.
  • The vertex is the point where the two rays meet to form the angle.

Types of angles

There are six types of angles and these types are identified by their measures. They are acute angle, right angle, obtuse angle, straight angle, reflex angle, and complete angle.

Acute angle

Acute angles are angles that are below 90° and above 0°.

Right angles

Right angles are angles that are exactly 90°.

Obtuse angles

Obtuse angles are angles that measure above 90° and measure below 180°.

Straight angles

Straight angles are angles that measure exactly 180°.

Below is a straight line. It means that the arms of the angles are opposite to each other.

Reflex angles

Reflex angles are angles that measure above 180° and measure below 360°.

Complete angles

Complete angles are angles that measure 360°.

What type of angle is it when it is 130°?

Solution

An obtuse angle. It is between 90° and 180° .

What is the type of angle in the diagram below?

Solution

Acute angle. It is below 90° .

What type of angle is it when it is 360°?

Solution

Complete angle. It is exactly 360°.

Angle measurement

Protractors are used to measure angles. Just like most quantities, angles require a unit to be measured in. These are in degrees, radians, and gradians.

Degree of an angle

Degrees are denoted by the symbol °. An angle is said to be 1° if its rotation from the reference goes as much as . By this we can establish that complete angles around a point are 360° always.

So right angles for example, are a of rotation around a point, and that's why a 90 degrees angle represents a quarter of a circle.

Radian of an angle

The radian measurement of angles is usually common in calculus. This is the SI unit of angles. It is denoted by 'rad'. The rotation around a point (complete angle) measures radians. This means that and .

Gradian of an angle

This unit of angles is the least commonly used in mathematics. It is denoted by 'grad'. The rotation around a point (complete angle) measures 400 grad.

This also means that of a full rotation. This also extensively means that a straight angle equals 200 grad.

Angle Rules

Angles possess certain properties with regard to parallel and intersecting lines. These properties are what help us calculate unknown angles. We will present these properties in the figure below.

  • . This means that vertically opposite angles are equal.
  • . The sum of adjacent angles such as a and b is 180°.
  • . Corresponding angles are also equal.
  • . The sum of the interior angle is 180°.
  • . Alternate angles are always equal.

Find angles a and b in the figure below.

Solution

The sum of adjacent angles is 180°, thus we have

Next, we isolate , to get

Alternate angles are always equal. Hence, , and thus

Find the angle labelled 5 in the figure below.

Solution

There are quite a number of ways to solve this using the different angle properties of lines. However, we can decide to look at the vertically opposite angle of 125°. We know that vertically opposite angles are equal, therefore,

We can now explore a different property of interior angles. The law says that interior angles sum up to 180°. Since angle 4 and angle 5 are interior angles in the diagram, we can now find angle 5.

Let us subtract 125° from each side of the equation to isolate angle 5.

Find the angle labeled 2 in the diagram below.

Solution

Angle 2 is a corresponding angle to 60°. According to the angle properties of lines, these two angles are equal, thus

.

Angles - Key takeaways

  • An angle is defined as the figure that is formed when two rays meet at a common point.
  • Angles are denoted by ∠ or commonly by Greek letters such as θ.
  • The types of angles are acute angle, right angle, obtuse angle, straight angle, reflex angle, and complete angle.
  • Angles are measured in either degrees, radians, or gradians.
  • Complete angles around a point are 360° always.

Final Angles Quiz

Question

If  , what is the value of angle 


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Answer


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Question

What is an angle?

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Answer

An angle is defined as the figure that is formed when two rays meet at a common point.

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Question

What is the symbol " " used to denote?

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Answer

Angles

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Question

The point where the two rays meet to form the angle is known as the ___


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Answer

Vertex

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Question

What are the arms of an angle?

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Answer

The arms of the angle are the two rays that come together to form the angle.

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Question

What type of angle is the 49° angle?

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Answer

Acute angle

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Question

What type of angle is exactly 90° 

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Answer

Right angle

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Question

What type of angle is the 158° angle?

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Answer

Obtuse angle

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Question

What is a complete angle?

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Answer

Complete angles are angles that measure 360°. 

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Question

Protractors are used to measure angles

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Answer

True

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Question

Angles are measured in all the following except ___

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Answer

Greek letters

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Question

Degrees are denoted by the symbol ___


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Answer

°

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Question

Which of the statements below is false

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Answer

Adjacent angles are equal

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Question

Find the angle labelled 3

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Answer

60° 

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Question

Find the angle labelled 8

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Answer

120° 

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Question

Find the angle labelled 7

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Answer

55° 

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