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.

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 co-interior angles.

• Alternate Interior Angles: Alternate interior 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: Co-interior angles are the pair of non-adjacent interior angles that lie 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.

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:

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°.

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 some regular polygons are calculated as follows:

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.

The relation between the co-interior angles can be determined by the co-interior angle theorem. As per this 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.

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 co-interior angles ∠1 and ∠4, and ∠2 and ∠3 are supplementary. Hence, the co-interior 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 co-interior angles are supplementary, then the lines are parallel. Hence, the co-interior 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(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.

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 S = 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 S = 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 S = is 180(n-2)° = 180(5-2) = 180 × 3 = 540°. Thus, the sum of the interior angles of a pentagon is 540°.