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.
What are Interior Angles?
In geometry, interior angles are formed in two ways. One is inside a polygon, and the other is when parallel lines cut by a transversal. 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.
- Interior Angles of Parallel 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 co-interior angles.
- Alternate Interior Angles: These angles are formed when two parallel lines are intersected by a transversal. This non-adjacent 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.
- Co-Interior Angles: These angles are the pair of non-adjacent interior angles on the same side of the transversal. In the above figure (b), the pairs of co-interior angles are ∠1 and ∠4, ∠2 and ∠3. These angles are also called same-side interior angles, or consecutive interior angles. The sum of two co-interior angles is 180º, that's why they form a pair of supplementary angles too.
Interior Angles of a Triangle
In a triangle, there are three interior angles at each vertex. The sum of those interior angles is always 180°. The bisectors of these angles meet at an point known as incenter. As the sum of interior angles of a triangle is 180°, there is only one possible right angle or obtuse angle possible in each triangle. A triangle with all three acute interior angles is called an acute triangle, a triangle with one interior angle as obtuse is known as an obtuse triangle, while a triangle with one interior angle as right angle is known as a right angled triangle.
Sum of Interior Angles Formula
From the simplest polygon, let us say a triangle, to an 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 polygons are calculated as follows:
|
Polygon |
Number of sides, n |
Sum of Interior Angles, S |
|---|---|---|
| Triangle | 3 | 180(3-2) = 180° |
| Quadrilateral | 4 | 180(4-2) = 360° |
| 5 | 180(5-2) = 540° | |
| 6 | 180(6-2) = 720° | |
|
Heptagon |
7 | 180(7-2) = 900° |
| 8 | 180(8-2) = 1080° | |
| Nonagon | 9 | 180(9-2) = 1260° |
| Decagon | 10 | 180(10-2) = 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°.
Interior Angles of 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(n-2)°. 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(n-2))/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(n-2))/n)° = ((180(5-2))/5)°
= (180 × 3)/5 = 540/5
= 108°
Thus, each interior angle of a regular pentagon = 108°.
Using the same formula, the interior angles of polygons are calculated as follows:
|
Regular Polygon |
Sum of Interior Angles, S |
Measurement of each interior angle((180(n-2))/n)° |
|---|---|---|
| Triangle | 180(3-2) = 180° | 180/3 = 60°, Here n = 3 |
| Square | 180(4-2) = 360° | 360/4 = 90°, Here n = 4 |
|
Pentagon |
180(5-2) = 540° | 540/5 = 108°, Here n = 5 |
|
Hexagon |
180(6-2) = 720° | 720/6 = 120°, Here n = 6 |
|
Heptagon |
180(7-2) = 900° | 900/7 = 128.57°, Here n = 7 |
|
Octagon |
180(8-2) = 1080° | 1080/8 = 135°, Here n = 8 |
| Nonagon | 180(9-2) = 1260° | 1260/9 = 140°, Here n = 9 |
| Decagon | 180(10-2) = 1440° | 1440/10 = 144°, Here n = 10 |
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(n-2)°.
- Each interior angle of a regular polygon of 'n' sides can be calculated using the formula ((180(n-2))/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 co-interior angles theorem, if a transversal intersects two parallel lines, each pair of co-interior angles is supplementary (their sum is 180°). Conversely, if a transversal intersects two lines such that a pair of co-interior angles are supplementary, then the two lines are parallel.
Interior Angles 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(n-2)º.
= 180(6-2)
= 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
Now, let us 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 co-interior angles and hence, they are supplementary (by co-interior 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 are Interior Angles?
Interior angles are those that lie inside a polygon. For example, a triangle has 3 interior angles. The other way to define interior angles is "angles enclosed in the interior region of two parallel lines when intersected by a transversal are known as interior angles".
How to Find the Sum of Interior Angles?
The sum of interior angles can be found by using the formula 180(n-2)° where n is the number of sides in a polygon. For example, to find the sum of interior angles of a quadrilateral, we replace n by 4 in the formula. We will get 180(4-2)°= 360°.
What is the Sum of the Interior Angles of a Heptagon?
A heptagon is a polygon with 7 sides and 7 angles. The sum of all the interior angles of a heptagon is 180(7-2)°, which is equal to 900°. Therefore, the sum of interior angles of a heptagon is 900 degrees.
What is the Sum of the Measures of the Interior Angles of a 27-Gon?
The sum of measures of interior angles of a 27-gon is 180(27-2)°. It is equal to 180 × 25, which is 4500°.
How to Solve Same Side Interior Angles?
Same side interior angles are supplementary when two parallel line are cut by a transversal. It means their sum is 180 degrees. So, to solve such angles, we will use this property and find the missing value.
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(n-2)°. 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(n-2)°, where n is the number of sides in a polygon. Here, n is 3 as the triangle has 3 sides. Hence, sum is 180(n-2)° = 180(3-2) = 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(n-2)°, where n is the number of sides in a polygon. Here, n is 6 as the hexagon has 6 sides. Hence, sum is 180(n-2)° = 180(6-2) = 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(n-2)°, where n is the number of sides in a polygon. Here, n is 5 as the pentagon has 5 sides. Hence, sum is 180(n-2)° = 180(5-2) = 180 × 3 = 540°. Thus, the sum of the interior angles of a pentagon is 540°.