Interior Angles

Interior Angles

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What are Interior Angles?

We can define interior angles in two ways.

1. The angles that lie inside a shape (generally a polygon) are said to be interior angles.

Example:

Interior angles examples

In the above figure, the angles \(a, b\) and \(c\) are interior angles. 

The angles \(d, e\) and \(f\) are called exterior angles.

2. The angles that lie in the area enclosed between two parallel lines that are intersected by a transversal are also called interior angles.

Example:

Interior angles of parallel lines intersected by a transversal

In the above figure, \(L_1\) and \(L_2\) are parallel and \(L\) is the transversal.

Here, the angles 1, 2, 3 and 4 are interior angles.

Alternate Interior Angles

Alternate interior angles are the pair of non-adjacent interior angles that lie on the opposite sides of the transversal.

In the above figure, the pairs of alternate interior angles are:

  • 1 and 3
     
  • 2 and 4

Co-Interior Angles

Co-interior angles are the pair of non-adjacent interior angles that lie on the same side of the transversal.

In the above figure, the pairs of co-interior angles are:

  • 1 and 4
     
  • 2 and 3

Sum of Interior Angles Formula (with Illustration)

We know that the sum of all the three interior angles of a triangle is 180\(^\circ\)

Sum of interior angles of a triangle is 180 degrees

We also know that the sum of all the four interior angles of any quadrilateral is 360\(^\circ\)

Sum of interior angles of a quadrilateral is 360 degrees

But what is the sum of the interior angles of a pentagon, hexagon, heptagon, etc?

Don't you think it would have been easier if there was a formula to find the sum of the interior angles of any polygon?

Well, your wish is granted!

This is the formula to find the sum of the interior angles of a polygon of \(n\) sides:

Sum of interior angles = 180(n-2)\(^\circ\)

Using this formula, let us calculate the sum of the interior angles of some polygons.

Polygon

Number of sides, \(\mathbf{ n}\) Sum of Interior Angles

Pentagon

\(5\) \(180(5-2) = \mathbf{540^\circ}\)

Hexagon

\(6\) \(180(6-2) = \mathbf{720^\circ}\)

Heptagon

\(7\) \(180(7-2) = \mathbf{900^\circ}\)

Octagon

\(8\) \(180(8-2) = \mathbf{1080^\circ}\)

You can observe this visually using the following illustration.

You can choose a polygon and drag its vertices.

You can then observe that the sum of all the interior angles in a polygon is always constant.


Finding an Unknown Interior Angle

We can find an unknown interior angle of a polygon using the "Sum of Interior Angles Formula".

For example:

Let us find the missing angle \(x^\circ\) in the following hexagon.

Finding an Unknown Interior Angle

From the above table, the sum of the interior angles of a hexagon is 720\(^\circ\)

Two of the interior angles of the above hexagon are right angles.

Thus, we get the equation:

\[ 90 + 90 + 140+150+130+x=720\]

Let us solve this to find \(x\).

\[ \begin{align} 600 + x &= 720\\[0.2cm]x&=120 \end{align}\]

Thus, the missing interior angle is: 

\(x^\circ = 120^\circ\)


Finding the Interior Angles of Regular Polygons (with Illustration)

A regular polygon is a polygon that has equal sides and equal angles.

Here are some examples of regular polygons:

Finding the interior angles of regular polygons

We already know that the formula for the sum of the interior angles of a polygon of \(n\) sides is \(180(n-2)^\circ\)

There are \(n\) angles in a regular polygon with \(n\) sides/vertices.

Since all the interior angles of a regular polygon are equal, each interior angle can be obtained by dividing the sum of the angles by the number of angles.

i.e.,

Each Interior Angle = \(\mathbf{\left(\dfrac{180(n-2)}{n} \right)^\circ}\)

Let us apply this formula to find the interior angle of a regular pentagon.

We know that the number of sides of a pentagon is \(n=5\)

Each interior angle of a regular pentagon can be found using the formula:

\[  \left(\!\dfrac{ 180(n-2)}{n} \!\right)^\circ \!\!=\!\!  \left(\!\dfrac{ 180(5-2)}{5} \!\right)^\circ\!\!=\!\!108^\circ\]

Thus, a regular pentagon will look like this:

Finding interior angles of a regular polygon: The interior angle of a regular pentagon is 108 degrees.

Would you like to see the interior angles of different types of regular polygons?

Here is an illustration for you to try.

You can move the slider to select the number of sides in the polygon and then click on "Go".


Alternate Interior Angle Theorem (with Illustration)

Suppose two parallel lines are intersected by a transversal, as shown below:

Alternate interior angles

What is the relation between any pair of alternate interior angles?

This relation is determined by the "Alternate Interior Angle Theorem"

Alternate Interior Angle Theorem

When a transversal intersects two parallel lines, each pair of alternate interior angles are equal.

Conversely, if a transversal intersects two lines such that a pair of interior angles are equal, then the two lines are parallel.

Proof :

Refer to the figure above.

We have:

\[ \begin{align} \angle 1 &= \angle 5 \text{ (corresponding angles)} \\[0.3cm] \angle 3 &= \angle 5 \text{ (vertically opposite angles)} \end{align} \]

Thus,

\[\angle 1 = \angle 3\]

Similarly, we can prove that \(\angle 2\) = \(\angle4\)

Proof of Converse

Conversely, suppose that

\[ \begin{align}\angle 1&= \angle 3 & \rightarrow (1) \end{align}\]

We have to prove that the lines are parallel.

As \(\angle 3 \) and \(\angle 5\) are vertically opposite angles,

\[ \begin{align}\angle 3 & = \angle 5 & \rightarrow (2) \end{align} \]

From (1) and (2),

\[\angle 1 = \angle 5\]

Thus, a pair of corresponding angles is equal, which can only happen if the two lines are parallel.

Hence, the alternate interior angle theorem is proved.

Would you like to observe visually how the alternate interior angles are equal?

Here is an illustration for you to test the above theorem. 

You can change the angles by moving the "Red" dot.

Choose "1st Pair" (or) "2nd Pair" and click on "Go".


Co-Interior Angle Theorem (with Illustration)

What about any pair of co-interior angles?

The relation between the co-interior angles is determined by the co-interior angle theorem.

Co-Interior Angle Theorem

If a transversal intersects two parallel lines, each pair of co-interior angles are supplementary (their sum is 180\(^\circ\))

Conversely, if a transversal intersects two lines such that a pair of co-interior angles are supplementary, then the two lines are parallel.

Proof :

Refer to the following figure once again: 

Co-interior angles

We have:

\[ \begin{align} \angle 1& = \angle 5 \;\;\;\text{ (corresponding angles)} \\[0.3cm]\angle 5 + \angle4& = 180^\circ \;\text{ (linear pair)}\end{align} \]

From the above two equations, \[\angle 1 + \angle4 = 180^\circ\]

Similarly, we can show that \[\angle 2 + \angle 3 = 180^\circ \]

Converse:

Conversely, let us assume that

\[ \begin{align}\angle 1 + \angle4 &= 180^\circ & \rightarrow (1) \end{align}\]

Since \(\angle 5\) and \(\angle 4\) forms linear pair,

\[ \begin{align}\angle 5 + \angle4 &= 180^\circ & \rightarrow (2) \end{align}\]

From (1) and (2),

\[ \angle 1 = \angle 5\]

Thus, a pair of corresponding angles is equal, which can only happen if the two lines are parallel.

Hence, the co-interior angle theorem is proved.

Would you like to observe visually how the co-interior angles are supplementary?

Here is an illustration for you to try.

You can change the angles by clicking on the purple point and click on "Go".

 
important notes to remember
Important Notes
  1. The sum of the interior angles of a polygon of \(n\) sides is \(\mathbf{180(n-2)^\circ}\)
  2. Each interior angle of a regular polygon of \(n\) sides is \(\mathbf{\left(\dfrac{180(n-2)}{n} \right)^\circ}\)
  3. Each pair of alternate interior angles is equal
  4. Each pair of co-interior angles is supplementary

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Solved Examples

Example 1

 

 

Find the interior angle at the vertex \(B\) in the following figure.

Interior angles solved example: Finding missing interior angle

Solution:

The number of sides of the given polygon is,

\(n=6\)

Thus, the sum of the interior angles of this polygon is:

\[ 180(n-2)=180(6-2)=720^\circ\]

We know that the sum of all the interior angles in this polygon is equal to 720 degrees.

The sum of all the angles of the given polygon is:

\[\begin{align} &\angle A+ \angle B +\angle C + \angle D + \angle E + \angle F\\[0.3cm] \!\!\!&\!\!=(x\!\!-\!\!60)\!+\!(x\!\!-\!\!20)\!+\!130\!+\!120\!+\!110\!+\!(x\!\!-\!\!40) \\[0.3cm]&=3x+240\end{align}\]

Now we set this sum equal to 720 and solve it for \(x\).

\[ \begin{align} 3x+240&=720\\[0.3cm] 3x &=480\\[0.3cm] x &=160 \end{align}\]

We have to find \(\angle B\).

\[\angle B = (x-20)^\circ = (160-20)^\circ = 140^\circ\]

\(\therefore\) \(\angle B = 140^\circ\)
Example 2

 

 

In the following figure, \(M N \| O P\) and \(O N \| P Q\).

If \(\angle M N O=55^\circ\) then find \(\angle O P Q\).

Co-interior angles example

Solution:

We will extend the lines in the given figure.

Co-interior angles example problem

Here, \(M N \| O P\) and \(ON\) is a transversal.

Thus, \(55^\circ\) and \(x\) are co-interior angles and hence, they are supplementary (by co-interior angle theorem). i.e.,

\[ \begin{align}55^\circ+x&=180^\circ\\[0.3cm] x &=125^\circ \end{align}\]

Again, \(O N \| P Q\) and \(OP\) is a transversal. 

Thus, \(x\) and \(\angle O P Q\) are corresponding angles and hence they are equal. i.e.,

\[ \angle O P Q = x = 125^\circ\]

\(\therefore\) \(\angle O P Q=125^\circ\)
Example 3

 

 

In the following figure, \(l \| m\) and \(s \| t\).

Find the value of \(x+y-z\)

Interior angles example problem: Find the value of x+y-z

Solution:

Since \(l \| m\) and \(t\) is a transversal, \(y^\circ\) and \(70^\circ\) are alternate interior angles.

Hence they are equal in measure (by alternate interior angle theorem). i.e.,

\[y^\circ =70^\circ\]

Again, \(s \| t\) and \(m\) is a transveral, \(x^\circ\) and \(70^\circ\) are the corresponding angles and hence they are equal. i.e., 

\[ x^\circ =70^\circ\]

Now let us assume that the angle that is adjacent to \(x^\circ\) is \(w^\circ\).

Alternate interior angles example

Since \(x^\circ\) and \(w^\circ\) form a linear pair,

\[ \begin{align} x^\circ + w^\circ &= 180^\circ\\[0.3cm] 70^\circ+w^\circ &=180^\circ\\[0.3cm]\\ w^\circ &= 110^\circ \end{align} \]

Now \(w^\circ\) and \(z^\circ\) are corresponding angles and hence, they are equal. i.e.,

\[z^\circ = w^\circ =110^\circ\]

Now,

\[x+y-z=70+70-110 = 30\]

\(\therefore\) \(x+y-z=30\)
 
Challenge your math skills
Challenging Questions
  1. In the following figure, \(\mathrm{AB}\|\mathrm{CD}\| \mathrm{EF}\)
    Interior angles challenging question
    Find the value of \(x\)

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Practice Questions

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

 
 
 
 
 
 

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. Do interior angles add up to 180\(^\circ\)?

Only the sum of co-interior angles is 180\(^\circ\).

2. What is the sum of the interior angles of a polygon?

The sum of the interior angles of a polygon of n sides is 180(n-2)\(^\circ\).

3. How do you find the interior angle?

Each interior angle of a regular polygon of n sides is \(\mathbf{\left(\dfrac{180(n-2)}{n} \right)^\circ}\)

 
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