Angles

Table of Contents 


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Introduction to Angles

When two lines intersect at a point, the measure of the “opening” between these two lines is called an “Angle”. It is denoted using the symbol \({\angle}\)

two lines intersect at a point to form an angle

Angles are usually measured in degrees and denoted by \({\circ}\) (the degree symbol), which is a measure of circularity or rotation.

Angles are a part of our day to day life. Engineers and architects use angles for the design of roads, buildings, and sporting facilities.

a surveyor using a Theodolite at a construction site to measure the angle

In the above image, we can see a surveyor using a Theodolite at a construction site to measure the angle.

Let’s see, how many of you like sports!

Have you ever watched a game of football?

Did you observe where the players take the corner kick from? Well, the point at which the lines intersect is what forms an angle!

angle formed at the corner kick mark in a football game

Parts of an Angle

In plane geometry, an angle is formed when two rays originate from the same endpoint. There are two parts of the angle-

  1. two rays, which are called the sides of the angle
  2. the vertex of the angle, which has a common endpoint that is shared by the two rays

the parts of an angle consist of two rays and a vertex

Angles and Degrees

An angle is measured in degrees. There are 360 degrees in one Full Rotation (one complete circle around).

An angle is measured in degrees 

Size of an Angle

The best way to measure the size of an angle is to use a protractor. The standard size of a protractor is \({180^\circ}\).

There are two sets of numbers on a protractor:

  • one in a clockwise direction
  • another in an anti-clockwise direction

a protractor is used to measure the size of an angle


Types of Angles and their Properties

  Type of Angle Description Example

 1.

Acute Angle

An angle that is less than \({90^\circ}\)

an acute angle is less than 90 degrees

2.

Right Angle

An angle that is exactly \({90^\circ}\)

a right angle is exactly 90 degrees

3.

Obtuse Angle

An angle that is greater than \({90^\circ}\) and less than \({180^\circ}\)

an obtuse angle is more than 90 degrees but less than 180 degrees

4.

Straight Angle

An angle that is exactly \({180^\circ}\)

a straight angle is exactly 180 degrees

5.

Reflex Angle

An angle that is greater than \({180^\circ}\) and less than \({360^\circ}\)

a reflex angle is more than 180 degrees but less than 360 degrees

6.

Complete Angle

An angle that is exactly \({360^\circ}\)

a complete angle is exactly 360 degrees

Acute Angle

An acute angle is an angle whose measure is greater than \({0^\circ}\) and less than \({90^\circ}\).

different types of acute angles

Right Angle

An angle that measures exactly \({90^\circ}\) is called a right angle.

It is generally formed when two lines are perpendicular to each other.

different types of right angles

Obtuse Angle

An obtuse angle is an angle whose measure is greater than \({90^\circ}\) and less than \({180^\circ}\).

different types of obtuse angles

Straight Angle

The angle formed by a straight line is called a straight angle.

It is one-half of the whole turn of a circle. The measure of the straight angle is \({180^\circ}\).

type of straight angle

Reflex Angle

A reflex angle is an angle whose measure is greater than \({180^\circ}\) but less than \({360^\circ}\)

different types of reflex angles

Complete Angle

An angle whose measure is equal to \({360^\circ}\) is called a complete angle.

a complete angle measuring 360 degrees forms a circle

In this simulation, you can drag the green line around the circle to see the type of angle that is formed between the two lines.

Observe the name of the angle formed at different degrees.


How to measure an Angle?

We use protractors to measure angles.

See the image below. We can see \({\angle AOB}\)

steps to measure the angle of a given example of angle AOB using a protractor

Let’s try and see if we can find out what type of angle is \({\angle AOB}\)

Doesn't it look like an acute angle? This means that its measure is greater than \({0^\circ}\) and less than \({90^\circ}\).

Now let’s try to find its exact measure with the help of a protractor

Steps to measure \({\angle AOB}\)

Step 1

Align the protractor with the ray OB as shown below.

Align the protractor with the ray OB

Start reading from the \({0^\circ}\) mark on the bottom-right of the protractor.

Step 2

The number on the protractor that coincides with the second ray is the measure of the angle.

The number on the protractor that coincides with the second ray is the measure of the angle.

Step 3

Measure the angle using the number on the "lower arc" of the protractor.

Measure the angle using the number on the "lower arc" of the protractor.

\[\begin{align} \angle {AOB} = 37^\circ\end{align}  \]

Next, let us try to measure this angle \({\angle AOC}\).

steps to measure the angle of a given example of angle AOC using a protractor

Step 1

Measure the angle from the \({0^\circ}\) mark on the bottom-left.

Measure the angle from the 0 degree mark on the bottom-left.

Step 2

The number on the "top arc" of the protractor that coincides with OA is the measure of \({\angle AOC}\).

The number on the "top arc" of the protractor that coincides with OA is the measure of angle AOC

\[\begin{align} \angle {AOC} = 143^\circ\end{align}  \]

You can try measuring various angles in this simulation by following a few simple steps.

  1. Enter the degree value that you want to measure on the protractor.
  2. Now using the blue line, drag it along the protractor to measure the degree.
  3. A right or wrong answer message will appear near the degree you entered to check your answer.

 
Thinking out of the box
Think Tank
  1. Is \({30^\circ}\) the same as \({-30^\circ}\)?difference between a 30 degree angle and a minus 30 degree angle

An Angle and its Reflex Angle

An angle is the "space" between two rays meeting at a common endpoint.

Let’s take the letters A,B, and O to name the angle.

understanding angles using an example

understanding reflex angles using an example

The larger angle is a Reflex Angle, but the smaller angle is an Acute Angle

It is to avoid this particular ambiguity that we refer to the angle marked above (with the dotted line) as reflex \({\angle AOB}\).

Note: any \({\angle AOB}\) and its reflex \({\angle AOB}\) sum to \({360^\circ}\)

This property of angle and its reflex can be used to measure (and construct) angles that measure more than \({180^\circ}\) and less than \({360^\circ}\).


How to construct Angles?

We use a protractor to construct angles.

Let’s draw a \({50^\circ}\) angle.

Step 1

First, draw a ray OB and align the protractor with OB as shown.

construct an angle by drawing a ray OB and align the protactor with OB

Step 2

Place a point above the marking on the protractor that corresponds to \({50^\circ}\).

Place a point above the marking on the protractor that corresponds to 50 degree

Step 3

Remove the protractor and draw a ray beginning at O that passes through this point.

\({\angle AOB}\) is the required angle.

That is \({\angle AOB}\) = \({50^\circ}\)

Remove the protractor and draw a ray beginning at O that passes through this point

Step 4

If the ray extends in the other direction, we measure the angle from the \({0^\circ}\) mark on the bottom-left.

if the ray extend in other direction, measure the angle from the 0 degree mark on the bottom-left.

A ray extending in other direction

The above image shows how to draw a \({50^\circ}\) angle when the ray is pointing in another direction.

 
tips and tricks
Tips and Tricks
  1. Mistake: Complementary angles have to be next to each other

    Fact: When the sum of two angles is \({90^\circ}\), the angles are called complementary angles. These angles need not be necessarily next to each other as long as their sum is \({90^\circ}\)

  2. Mistake: Any angles next to each other are adjacent angles.

    Fact: For angles to be adjacent, they have to share a common side and vertex.

Interior and Exterior Angles

Interior of an Angle

Interior angle is an angle inside the shape. The area between the rays that makes up an angle, and extending away from the vertex to infinity. The sum of the interior angles of a triangle is always 180 degree.

Exterior of an Angle

The exterior angle is the angle between any side of a shape, and a line extended from the next side. The sum of an exterior angle and its adjacent interior angle is also 180 degree.

understanding the exterior of an angle


Complementary and Supplementary Angles

Complementary Angles

When the sum of two angles is \({90^\circ}\), the angles are called complementary angles.

Each angle is called the complement of the other angle.

sum of two complementary angles is 90 degrees

\({\angle AOB}\) + \({\angle BOC}\) = \({90^\circ}\)

example of complementary angles

\({60^\circ}\) + \({30^\circ}\) = \({90^\circ}\)

The above angles are complementary angles.

Supplementary Angles

When the sum of two angles is \({180^\circ}\), the angles are called supplementary angles.

Each angle is called the supplement of the other angle.

sum of two complementary angles is 180 degrees

\({\angle AOB}\) + \({\angle BOC}\) = \({180^\circ}\)

 

example of supplementary angles

\({130^\circ}\) + \({50^\circ}\) = \({180^\circ}\)

The above angles are supplementary angles


Some more types of Angles

Adjacent Angles

Any two angles that share-

  • a common ray or side
  • a common vertex
  • and whose interiors do not overlap

are called adjacent angles.

example of adjacent angles

Interiors of \({\angle ABD}\) and \({\angle CBD}\) don’t overlap and hence they are adjacent angles.

Vertically Opposite Angles

The angles opposite to each other when two lines cross are called vertically opposite angles. They are always equal.

example of vertically opposite angles

Example: \({\angle PTS}\) and \({\angle RTQ}\) are vertically opposite angles.


What is Transversal?

Consider two lines AB and CD

figure showing two horizontal lines AB and CD

Let line segment XY be the line that intersects these two lines at two distinct points, P and Q.

figure showing vertical line XY intersecting two other horizontal lines

A line that intersects two other lines at two distinct points is known as the transversal to the lines.

In the above image, XY is the transversal. When a transversal crosses a pair of parallel lines, you end up with different types of angles.

Let's discuss these angles.

Corresponding Angles

Corresponding angles are the angles that:

  • have different vertices
  • lie on the same side of the transversal, and lie above (or below) the lines
  • They are always equal.

examples of corresponding angles

When a transversal intersects two parallel lines, the corresponding angles formed are always equal.

In the above figure, \({\angle 1}\) & \({\angle 5}\), \({\angle 2}\) & \({\angle 6}\), \({\angle 4}\) & \({\angle 8}\), \({\angle 3}\) & \({\angle 7}\) are all pairs of corresponding angles.

Alternate Interior Angles

Alternate-interior angles are those angles that:

  • have different vertices
  • lie on the alternate sides of the transversal
  • lie between the interior of the two lines

examples of alternate interior angles

When a transversal intersects two parallel lines, the alternate interior angles formed are always equal.

Here, \({\angle 1}\) and \({\angle 2}\) are the alternate interior angles.

Alternate Exterior Angles

Alternate-exterior angles are those angles that:

  • have different vertices
  • lie on the alternate sides of the transversal
  • are exterior to the lines

examples of alternate exterior angles

 When a transversal intersects two parallel lines, alternate exterior angles formed are always equal.

Here, \({\angle 1}\) & \({\angle 2}\) have different vertices.

They lie on the opposite side of the transversal.

Therefore, \({\angle 1}\) & \({\angle 2}\) are alternate exterior angles.

Co-interior Angles

Co-interior angles are those angles that:

  • have different vertices
  • lie between two lines
  • and are on the same side of the transversal

examples of co-interior angles

When a transversal intersects two parallel lines, the co-interior angles are always supplementary.

In the above image, \({\angle 1}\) & \({\angle 2}\) have different vertices.

They lie on the same side of the transversal. 

\({\angle 1}\) & \({\angle 2}\) are angles interior to the lines AB and CD respectively.

Therefore, \({\angle 1}\) and \({\angle 2}\) are co-interior angles and they are supplementary.


Exterior Angle Property of a Triangle

An exterior angle of a triangle is always equal to the sum of the opposite interior angles.

Consider one of the exterior angles, \({\angle 4}\) of △PQR

exterior angle property of a triangle

In △PQR, by the angle sum property,

\begin{array}{l}
\angle 1+\angle 2+\angle 3=180^{\circ} \\
\angle 2+\angle 4=180^{\circ} \\
\angle 1+\angle 3=180^{\circ}-\angle 2\\
\angle 4=180^{\circ}-\angle 2
\end{array}


The sum obtained in both cases is equal, \({\angle 1}\) + \({\angle 3}\) = \({\angle 4}\)

In any triangle, an exterior angle is always equal to the sum of its interior opposite angles.

This is the exterior angle property of triangles.


Angle types based on Rotation

Based on the direction of measurement or the direction of rotation, angles can be of two types:

  1. Positive Angles
  2. Negative Angles

Positive Angles

An angle generated by the counterclockwise direction is a positive angle.

From the origin, if an angle is drawn in the \((+\textit {x} ,+\textit {y})\) plane, it forms a positive angle.

Negative Angles

Negative angles are those angles which are measured in a clockwise direction from the base.

From the origin, if an angle is drawn towards the \((-\textit {x} ,-\textit {y})\) plane, it forms a negative angle.

difference between negative angles and positive angles

 
important notes to remember
Important Notes
  1. \({0^\circ}\) < Acute angle < \({90^\circ}\)
  2. \({90^\circ}\) < Obtuse angle  < \({180^\circ}\)
  3. \({180^\circ}\) < Reflex angle < \({360^\circ}\)
  4. Right angle is an angle whose measure is \({90^\circ}\)
  5. Protractors usually have two sets of numbers going in opposite directions. When in doubt think “should this angle be bigger or smaller than \({90^\circ}\)       

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Solved Examples

Example 1



 

Determine the missing angle.

solved example to find the missing example

Solution:

Notice that the two angles form a right angle, when together.

This means that the angles are complementary and have a sum of \({90^\circ}\).

\[\begin{align}90^\circ - 62^\circ = 28^\circ\end{align}  \]

\(\textit {∴ The missing angle} = 28^\circ \)
Example 2



 

Determine the missing angle.

example to find a missing angle from the figure

Solution:

These two angles form a straight line. Straight lines measure \({180^\circ}\).

That means that these two angles are supplementary.

\[\begin{align}180^\circ - 77^\circ = 103^\circ\end{align}  \]

\(\textit {∴ The missing angle} = 103^\circ \)
Example 3



 

Find the angle marked \(\textit x \) in the picture below. BD and EG are parallel lines.

State which angle fact you used at each step.

example showing how to find a missing angle when a transversal interesects two lines

Solution:

Look at the figure, \({\angle CFG}\), and the missing angle \(\textit x \) are corresponding angles.

steps to finding the missing angle

\[\begin{align} \angle {CFE} + \angle {CFG} = 180^\circ\end{align}  \]

Therefore,

\[\begin{align} \angle {CFG} = 180^\circ\ - 32^\circ\ = 148^\circ\end{align}  \]

\(\textit {∴ The angle x} = 148^\circ \)

 

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Practice Questions

Here are few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Important Topics

Given below are the list of topics that are closely connected to angles. These topics will also give you a glimpse of how such concepts are covered in Cuemath.


Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. What are the 5 types of angles?

The 5 types of angles are: right angles, acute angles, obtuse angles, straight angles and reflex angles.

2. What is an angle in math?

In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. An angle is denoted using the symbol \(\angle\)

3. What are the 7 types of angles?

The 7 types of angles are: right angles, acute angles, obtuse angles, straight angles, reflex angles, full angles, and complementary angles.

4. How do you describe angles?

In plane geometry, an angle can be described as a figure formed by two rays meeting at a common endpoint called the vertex of the angle.

 
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