Angles
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 \({\angle}\)
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.
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!
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
 two rays, which are called the sides of the angle
 the vertex of the angle, which has a common endpoint that is shared by the two rays
Angles and Degrees
An angle is measured in degrees. There are 360 degrees in one Full Rotation (one complete circle around).
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 anticlockwise direction
Types of Angles and their Properties
Type of Angle  Description  Example  

1. 
Acute Angle 
An angle that is less than \({90^\circ}\) 

2. 
Right Angle 
An angle that is exactly \({90^\circ}\) 

3. 
Obtuse Angle 
An angle that is greater than \({90^\circ}\) and less than \({180^\circ}\) 

4. 
Straight Angle 
An angle that is exactly \({180^\circ}\) 

5. 
Reflex Angle 
An angle that is greater than \({180^\circ}\) and less than \({360^\circ}\) 

6. 
Complete Angle 
An angle that is exactly \({360^\circ}\) 
Acute Angle
An acute angle is an angle whose measure is greater than \({0^\circ}\) and less than \({90^\circ}\).
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.
Obtuse Angle
An obtuse angle is an angle whose measure is greater than \({90^\circ}\) and less than \({180^\circ}\).
Straight Angle
The angle formed by a straight line is called a straight angle.
It is onehalf of the whole turn of a circle. The measure of the straight angle is \({180^\circ}\).
Reflex Angle
A reflex angle is an angle whose measure is greater than \({180^\circ}\) but less than \({360^\circ}\)
Complete Angle
An angle whose measure is equal to \({360^\circ}\) is called a complete angle.
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 \({\angle AOB}\)
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.
Start reading from the \({0^\circ}\) mark on the bottomright of the protractor.
Step 2
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.
\[\begin{align} \angle {AOB} = 37^\circ\end{align} \] 
Next, let us try to measure this angle \({\angle AOC}\).
Step 1
Measure the angle from the \({0^\circ}\) mark on the bottomleft.
Step 2
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.
 Enter the degree value that you want to measure on the protractor.
 Now using the blue line, drag it along the protractor to measure the degree.
 A right or wrong answer message will appear near the degree you entered to check your answer.
 Is \({30^\circ}\) the same as \({30^\circ}\)?
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.
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.
Step 2
Place a point above the marking on the protractor that corresponds to \({50^\circ}\).
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 \({\angle AOB}\) = \({50^\circ}\)
Step 4
If the ray extends in the other direction, we measure the angle from the \({0^\circ}\) mark on the bottomleft.
The above image shows how to draw a \({50^\circ}\) angle when the ray is pointing in another direction.

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 \({90^\circ}\)
 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.
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.
\({\angle AOB}\) + \({\angle BOC}\) = \({90^\circ}\)
\({60^\circ}\) + \({30^\circ}\) = \({90^\circ}\)
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.
\({\angle AOB}\) + \({\angle BOC}\) = \({180^\circ}\)
\({130^\circ}\) + \({50^\circ}\) = \({180^\circ}\)
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.
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: \({\angle PTS}\) and \({\angle RTQ}\) are vertically opposite angles.
What is Transversal?
Consider two lines AB and CD
Let line segment XY be the line that intersects these two lines at two distinct points, P and Q.
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.
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
Alternateinterior angles are those angles that:
 have different vertices
 lie on the alternate sides of the transversal
 lie between the interior of the two lines
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
Alternateexterior angles are those angles that:
 have different vertices
 lie on the alternate sides of the transversal
 are exterior to the lines
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.
Cointerior Angles
Cointerior angles are those angles that:
 have different vertices
 lie between two lines
 and are on the same side of the transversal
When a transversal intersects two parallel lines, the cointerior 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 cointerior 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
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:
 Positive Angles
 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.
 \({0^\circ}\) < Acute angle < \({90^\circ}\)
 \({90^\circ}\) < Obtuse angle < \({180^\circ}\)
 \({180^\circ}\) < Reflex angle < \({360^\circ}\)
 Right angle is an angle whose measure is \({90^\circ}\)
 Protractors usually have two sets of numbers going in opposite directions. When in doubt think “should this angle be bigger or smaller than \({90^\circ}\)
Solved Examples
Example 1 
Determine the missing angle.
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.
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.
Solution:
Look at the figure, \({\angle CFG}\), and the missing angle \(\textit x \) are corresponding angles.
\[\begin{align} \angle {CFE} + \angle {CFG} = 180^\circ\end{align} \]
Therefore,
\[\begin{align} \angle {CFG} = 180^\circ\  32^\circ\ = 148^\circ\end{align} \]
\(\textit {∴ The angle x} = 148^\circ \) 
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.
 Pairs of Angles
 Transversals and Related Angles
 Corresponding Angles
 Interior Angles
 Lines Parallel To The Same Line
 Essence of Geometrical Constructions
 Constructing Angle Bisectors
 Constructing Perpendicular Bisectors
 Constructing An Angle of 90 Degrees
 Constructing An Angle of 60 Degrees
 Constructing Perpendicular From Point to Line
 Protractor
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.