Interior Angles
The angles that lie inside a shape, are said to be interior angles, or the angles that lie in the area bounded between two parallel lines that are intersected by a transversal are also called interior angles. Let's learn more about interior angles in this article.
Interior Angles and their Types
There are different types of angles in nature, where each one of them carries some significance in our daily lives. Engineers, architects make use of angles in designing buildings, bridges, machines, and roads. The concept of angles is applied in sports as athletes use them to improve their performance. Like how soccer players must use a certain angle in order to pass the ball to the next player. It is very essential for one to learn about the different types of angles. Primarily, angles are categorized into different types based on their measurements. There are other types of angles known as pair angles since they appear in pairs in order to exhibit a certain property. Interior angles are one such kind.
We can define interior angles in two ways:
 Angles inside a Polygon: The angles that lie inside a shape, generally a polygon, are said to be interior angles. In the below figure (a), the angles ∠a, ∠b, and ∠c are interior angles.
 Angles inside lines: The angles that lie in the area enclosed between two parallel lines that are intersected by a transversal are also called interior angles. In the below figure (b), \(L_1\) and \(L_2\) are parallel, and L is the transversal. The angles ∠1, ∠2, ∠3, and ∠4 are interior angles.
Types of Interior Angles
There are two types of interior angles formed when two straight lines are cut by a transversal, and those are alternate interior angles and cointerior angles.
 Alternate Interior Angles: Alternate interior angles are formed when two parallel lines are intersected by a transversal. This nonadjacent pair of angles are formed on the opposite sides of the transversal. In the above figure (b), the pairs of alternate interior angles are ∠1 and ∠3, ∠2 and ∠4. They are equal in measurement if two parallel lines are cut by a transversal.
 CoInterior Angles: Cointerior angles are the pair of nonadjacent interior angles that lie on the same side of the transversal. In the above figure (b), the pairs of cointerior angles are ∠1 and ∠4, ∠2 and ∠3. These angles are also called sameside interior angles, or consecutive interior angles. The sum of two cointerior angles is 180º, that's why they form a pair of supplementary angles too.
Sum of Interior Angles Formula
From the simplest polygon, let us say a triangle, to the infinitely complex polygon with n sides such as octagon, all the sides of polygon create a vertex, and that vertex has an interior and exterior angle. As per the angle sum theorem, the sum of all the three interior angles of a triangle is 180°. Multiplying two less than the number of sides times 180° gives us the sum of the interior angles in any polygon.
Sum, S = (n − 2) × 180°
Here, S = sum of interior angles and n = number of sides of the polygon
Applying this formula on a triangle, we get:
S = (n − 2) × 180°
S = (3 − 2) × 180°
S = 1 × 180°
S = 180°
Using the same formula, the sum of the interior angles of some more polygons are calculated as follows:
Polygon 
Number of sides, n  Sum of Interior Angles, S 

Triangle  3  180(32) = 180° 
Rectangle  4  180(42) = 360° 
5  180(52) = 540°  
6  180(62) = 720°  
Heptagon 
7  180(72) = 900° 
8  180(82) = 1080°  
Nonagon  9  180(92) = 1260° 
Decagon  10  180(102) = 1440° 
Finding an Unknown Interior Angle
We can find an unknown interior angle of a polygon using the "Sum of Interior Angles Formula". Let us consider the below example to find the missing angle ∠x in the following hexagon.
From the above given interior angles of a polygon table, the sum of the interior angles of a hexagon is 720°. 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
600 + x = 720
x = 720  600 = 120
Thus, the missing interior angle x is 120°.
Finding Interior Angles of Regular Polygons
A polygon can be considered as a regular polygon when all its sides and angles are congruent. Here are some examples of regular polygons:
We already know that the formula for the sum of the interior angles of a polygon of 'n' sides is 180(n2)°. 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 sides.
Each Interior Angle = ((180(n2))/n)°
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 5 (Here, n = 5).
Each interior angle of a regular pentagon can be found using the formula:
((180(n2))/n)° = ((180(52))/5)°
= (180 × 3)/5 = 540/5
= 108°
Thus, each interior angle of a regular pentagon = 108°.
Using the same formula, the interior angles of some regular polygons are calculated as follows:
Regular Polygon 
Sum of Interior Angles, S  Measurement of each interior angle((180(n2))/n)° 
Triangle  180(32) = 180°  180/3 = 60°, Here n = 3 
Rectangle  180(42) = 360°  360/4 = 90°, Here n = 4 
Pentagon 
180(52) = 540°  540/5 = 108°, Here n = 5 
Hexagon 
180(62) = 720°  720/6 = 120°, Here n = 6 
Heptagon 
180(72) = 900°  900/7 = 128.57°, Here n = 7 
Octagon 
180(82) = 1080°  1080/8 = 135°, Here n = 8 
Nonagon  180(92) = 1260°  1260/9 = 140°, Here n = 9 
Decagon  180(102) = 1440°  1440/10 = 144°, Here n = 10 
Alternate Interior Angle Theorem
The relation between any pair of alternate interior angles can be determined by using the alternate interior angle theorem. As per the 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 :
Suppose two parallel lines \(L_1\) and \(L_2\) are intersected by a transversal L, as shown below:
We have:
∠1 = ∠5 (Corresponding angles are congruent as per the corresponding angles theorem)
∠3 = ∠5 (Vertically Opposite angles are congruent as per the vertical angles theorem)
Thus, ∠1 = ∠3
Similarly, we can prove that ∠2 = ∠4
Thus, when the transversal L intersects the parallel lines \(L_1\) and \(L_2\), each pair of alternate interior angles ∠1 and ∠3, and ∠2 and ∠4 are equal. Hence, the alternate interior angle theorem is proved.
Proof of Converse
Conversely, suppose that ∠1 = ∠3 > (1)
We have to prove that the lines \(L_1\) and \(L_2\) are parallel.
As ∠3 and ∠5 are vertically opposite angles, ∠3 = ∠5 > (2)
From (1) and (2),
∠1 = ∠5
Thus, by using the converse of corresponding angles theorem, we can state that if two corresponding angles are equal, then the lines are parallel to each other. Hence, it is also proved that if two alternate interior angles are congruent, then the lines are parallel.
CoInterior Angle Theorem
The relation between the cointerior angles can be determined by the cointerior angle theorem. As per this theorem, if a transversal intersects two parallel lines, each pair of cointerior angles is supplementary (their sum is 180°). Conversely, if a transversal intersects two lines such that a pair of cointerior angles are supplementary, then the two lines are parallel.
Proof:
Suppose two parallel lines \(L_1\) and \(L_2\) are intersected by a transversal L, as shown in the above image. We have:
∠1 = ∠5 (Corresponding angles are congruent as per the corresponding angles theorem)
∠5 + ∠4 = 180° (Linear Pair of angles are supplementary as per the linear pair axiom)
From the above two equations, ∠1 + ∠4 = 180°
Similarly, we can show that ∠2 + ∠3 = 180°
Thus, when the transversal L intersects the parallel lines \(L_1\) and \(L_2\), each pair of cointerior angles ∠1 and ∠4, and ∠2 and ∠3 are supplementary. Hence, the cointerior angle theorem is proved.
Proof of Converse
Conversely, let us assume that ∠1 + ∠4 = 180° > (1)
Since ∠5 and ∠4 forms linear pair, ∠5 + ∠4 = 180° > (2)
From (1) and (2),
∠1 = ∠5
Thus, by using the converse of corresponding angles theorem, we can state that if two corresponding angles are equal, then the lines are parallel to each other. Hence, it is also proved that if two cointerior angles are supplementary, then the lines are parallel. Hence, the cointerior angle theorem converse is proved.
Related Articles on Interior Angles
Check out the following pages related to interior angles.
 Vertical Angles
 Alternate Angles
 Alternate Exterior Angles
 Same Side Interior Angles
 Interior Angles of Polygon Calculator
Important Notes
Here is a list of a few points that should be remembered while studying interior angles:
 The sum of the interior angles of a polygon of 'n' sides can be calculated using the formula 180(n2)°
 Each interior angle of a regular polygon of 'n' sides can be calculated using the formula ((180(n2))/n)°
 As per the alternate interior angles 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.
 As per the cointerior angles theorem, if a transversal intersects two parallel lines, each pair of cointerior angles is supplementary (their sum is 180°). Conversely, if a transversal intersects two lines such that a pair of cointerior angles are supplementary, then the two lines are parallel.
Interior Angles Solved Examples

Example 1: Find the interior angle at vertex B in the following figure.
Solution:
The number of sides of the given polygon is n =6, so it's a hexagon (Hexagon has 6 sides). Thus, the sum of the interior angles of this polygon is:
180(n2)º
= 180(62)
= 180 × 4 = 720°
We know that the sum of all the interior angles in this polygon is equal to 720°. The sum of all the angles of the given polygon is:
∠A + ∠B + ∠C + ∠D + ∠E + ∠F
= (x  60) + (x  20) + 110 + 120 + 130 + (x  40)
= 3x+ 240
Now we set this sum equal to 720 and solve it for x
3x+ 240 = 720
3x = 480
x = 480/3 = 160
We have to find ∠B
∠B = (x  20)° = (160  20)° = 140°
Therefore, The interior angle at vertex B is ∠B = 140°.

Example 2: In the following figure, MN  OP and ON  PQ. If ∠MNO=55°, then find ∠OPQ.
Solution:
We will extend the lines in the given figure.
Here, MN  OP, and ON is a transversal. Thus, 55° and x° are cointerior angles and hence, they are supplementary (by cointerior angle theorem). i.e.,
55° + x° = 180°
x = 180  55 = 125°
Again, ON  PQ and OP is a transversal. Thus, x° and ∠OPQ are corresponding angles and hence they are equal. i.e.,
∠OPQ = x = 125°
Therefore, ∠OPQ = 125°
FAQs on Interior Angles
What is the Sum of the Interior Angles of a Polygon?
The sum of the interior angles of a polygon of n sides can be calculated with the formula 180(n2)°. It helps us in finding the total sum of all the angles of a polygon, whether it is a regular polygon or an irregular polygon. By using this formula, we can verify the angle sum property as well. The sum of all the interior angles of a triangle is 180º, the interior angle sum of a quadrilateral is 360º, and so on.
What is the Sum of the Interior Angles of a Triangle?
Let's calculate the sum of the interior angles of a triangle using the sum of interior angles formula S = 180(n2)°, where n is the number of sides in a polygon. Here, n is 3 as the triangle has 3 sides. Hence, sum S = is 180(n2)° = 180(32) = 180°. Thus, the sum of the interior angles of a triangle is 180°.
What is the Sum of Interior Angles of a Hexagon?
Let's calculate the sum of the interior angles of a hexagon, using the sum of interior angles formula S = 180(n2)°, where n is the number of sides in a polygon. Here, n is 6 as the hexagon has 6 sides. Hence, sum S = is 180(n2)° = 180(62) = 180 × 4 = 720°. Thus, the sum of the interior angles of a hexagon is 720°.
How Many Interior Angles Does an Octagon Have?
An octagon has eight sides and thus, it has eight interior angles. The sum of those eight interior angles of an octagon is 1080º.
What is the Sum of all Interior Angles of a Pentagon?
Let's calculate the sum of the interior angles of a pentagon, using the sum of interior angles formula S = 180(n2)°, where n is the number of sides in a polygon. Here, n is 5 as the pentagon has 5 sides. Hence, sum S = is 180(n2)° = 180(52) = 180 × 3 = 540°. Thus, the sum of the interior angles of a pentagon is 540°.