Exterior Angle Theorem

Exterior Angle Theorem is one of the foundational theorems in geometry, as it describes the relationship between exterior and interior angles in any triangle. An Exterior Angle is formed when any side of a polygon is extended to one side. In simple terms, an Exterior Angle is an angle that exists outside the boundaries of the polygon but shares a vertex with the polygon.

This article explores the relationship between the exterior angle and the remote angle of a triangle, i.e., the Exterior Angle Theorem. We will cover various topics related to the Exterior Angle Theorem, including its statement, proof, and some applications as well. Also, we will learn the Exterior Angle Inequality Theorem, as it is somewhat related to the Exterior Angle Theorem.

What is Exterior Angle?

The angle formed between a side of the polygon and by extending the adjacent side is called the exterior angle of the polygon. The exterior angle and its adjacent angle follow the linear property i.e., the sum of the exterior angle and its adjacent angle is 180 degrees.

What is Exterior Angle in Triangle?

The angle formed by extending one side of the triangle and its adjacent side is called as exterior angle of the triangle. In other words, the angle formed outside the triangle is known as Exterior Angle in Triangle.

There are six exterior angles in a triangle. The exterior angle and the adjacent interior angle form a linear pair of angles i.e., the sum of exterior and corresponding interior angle is 180°.

In the above figure, angle 1, 2 and 3 are interior angles and angle 4 is exterior angle of the triangle.

What is Exterior Angle Theorem?

The exterior angle theorem states that:

When a side of the triangle is extended, then the exterior angle formed is equal to the sum of two opposite interior angles.

In the above figure of triangle ABC and BC be the extended side where, ∠ACD is the exterior angle of the triangle and ∠ABC and ∠BAC are the two opposite interior angles. Then, by exterior angle theorem:

∠ACD = ∠ABC + ∠BAC

Proof of Exterior Angle Theorem

Consider a triangle PQR and QR be the extended side of the triangle. A line RS is drawn parallel to side PQ of the triangle as shown in the following figure.

PQ || RS and PR is the transversal and ∠RPQ and ∠PRS are pair of alternate interior angles.

∠RPQ = ∠PRS

⇒∠1 = ∠x

PQ || RS and QT is the transversal and ∠PQR and ∠SRT are corresponding angles.

∠PQR = ∠SRT

⇒ ∠2 = ∠y

From above statements

∠1 + ∠2 = ∠x + ∠y

From above figure

∠x + ∠y = ∠PRT

From above two statements

∠1 + ∠2 = ∠PRT

⇒ ∠QPR + ∠PQR = ∠PRT [Exterior angle theorem]

From the above table, the exterior angle ∠PRT is equal to the sum of the two opposite interior angles ∠QPR and ∠PQR in triangle PQR.

Exterior Angle Inequality Theorem

The exterior angle inequality theorem states that:

The value of the exterior angle of a triangle is always greater than the value of either of the opposite angles of the triangle.

In the above figure the ∠1, ∠2 and ∠3 are interior angles of triangle ABC and ∠4 is exterior angle of the triangle ABC. The interior angles ∠1 and ∠2 is the opposite interior angles w.r.t exterior angle ∠4.

According to the Exterior Angle Inequality Theorem, the exterior angle ∠4 is greater than the either of the opposite interior angles ∠1 and ∠2.

∠4 > ∠1 and ∠4 > ∠2

Interior and Exterior Angles in a Triangle

As the name suggests, interior and exterior angles refer to the angles in a triangle that are either inside or outside the geometric shape. Apart from this, there are some more differences between these two types of angles in a triangle, and these differences are listed in the following table:

Type of Angle	Definition	Measurement
Interior Angle	An angle formed inside a triangle between two sides	Sum of all interior angles in a triangle is always 180°.  For an equilateral triangle, each angle in a triangle is 180° divided by 3 i.e., 60 ° or varying in other types of triangles.
Exterior Angle	An angle formed outside a triangle when one side is extended	Sum of an exterior angle and the adjacent interior angle is always 180 °.  The measurement of an exterior angle is equal to the sum of the measurements of the two non-adjacent interior angles.

Applications of the Exterior Angle Theorem

The applications of the exterior angle theorem are:

• To Find the Unknown Exterior Angles of Triangle
• To Find the Unknown Interior Angles of Triangle

Let's discuss these cases in detail with the help of examples.

How to Find the Unknown Exterior Angles of Triangle?

To find the unknown exterior angle of the triangle we use the exterior angle theorem. If the value of two remote interior angles are given we can easily find the value of exterior angles.

Example: Given a triangle with an exterior angle and remote interior angles. The value of the two remote interior angles are 30° and 65°. Find the value of the exterior angle.

Solution:

By the exterior angle theorem

Exterior angle = sum of remote interior angles. ∠e = ∠i1 + ∠i2

⇒ The value of exterior angle = 30° + 65°

⇒ The value of exterior angle = 95°

How to Find the Unknown Interior Angles of Triangle?

To find the unknown exterior angle of the triangle we use the exterior angle theorem. If the value of two remote interior angles are given we can easily find the value of exterior angles.

Example: Given a triangle with an exterior angle and remote interior angles. The value of the one of the remote interior angles and exterior angle is are 80° and 165°. Find the value of the other remote interior angle.

Solution:

By the exterior angle theorem

Exterior angle = Sum of Remote Interior Angles.

⇒ ∠e = ∠i₁ + ∠i₂

⇒ 165° = 80° + ∠i₂

⇒ ∠i₂ = 165° - 80°

The value of other interior angle = 85°

Read More,

• Triangles in Geometry
• Types of Triangles
• Area of Triangle
• Triangle Inequality Theorem
• Congruence of Triangles
• Types of Angles

Solved Examples on Exterior Angle Theorem

Example 1: Find the value of the exterior angle from the below figure:

Solution:

By the exterior angle theorem

∠ABC + ∠BAC = ∠ACD

⇒ e = 50° + 67°

⇒ e = 117°

Example 2: Find the value of i using the information from the following figure:

Solution:

By the exterior angle theorem

∠ABC + ∠BAC = ∠ACD

⇒ 40° + i = 170°

⇒ i = 170° - 40°

⇒ i = 130°

Example 3: Find the value of x

Solution:

By exterior angle theorem

∠ABC + ∠BAC = ∠ACD

⇒ 4x + 6x = 110°

⇒ 10x = 110°

⇒ x = 11°

Example 4: Find the value of x

Solution:

Since the exterior angle and adjacent interior has linear property

∠ACB + ∠ACD = 180°

⇒ x + 130° = 180°

⇒ x = 180°- 130°

⇒ x = 50°

Example 5: Find the values of interior angles.

Solution:

By the exterior angle theorem

∠ACD = ∠A + ∠B

⇒ 130 = y + 2 + y + 10

⇒ 2y + 12 = 130

⇒ 2y = 118

⇒ y = 59°

The interior angles are 61° and 69°

Practice Problems on Exterior Angle Theorem

Problem 1: In a triangle, one of the exterior angles measures 110 degrees and both remote angles are equal. What are the measures of the two remote interior angles?

Problem 2: If the measure of one of the remote interior angles of a triangle is 45° and other is 70° then what is the measure of the corresponding exterior angle?

Problem 3: In a triangle, the measure of all exterior angles is 120°. Find the measures of the interior angles.

Problem 4: A triangle has exterior angles measuring 70°, 80°, and 110°. Find the measures of the three corresponding interior angles.

Problem 5: Two angles of a triangle measure 40° and 65°. Find the measure of the third angle and corresponding Exterior Angle as well.

Problem 6: If an exterior angle of a triangle is 80^∘and one of the interior angles is 40^∘ find the measure of the third interior angle.

Problem 7: In a triangle, the exterior angle at vertex C is 120^∘. If the interior angles at A and B are 45^∘ and 15^∘ respectively determine if the triangle is valid.

Problem 8: A triangle has an exterior angle measuring 150^∘. Determine the measures of the two interior angles of the triangle.

Problem 9: If an exterior angle of a triangle is 105^∘ and the measure of one of the interior angles is 35^∘ find the measure of the remaining interior angle.

Problem 10: In a triangle, the exterior angle at vertex A is 130^∘. Find the measure of the other two interior angles of the triangle.

Conclusion

The Exterior Angle Theorem states that an exterior angle of the triangle is equal to the sum of the two remote interior angles. This theorem is fundamental in the understanding triangle properties and is used to the solve various geometric problems involving the triangles. By applying this theorem we can easily find unknown angles in the triangle verify the validity of triangles and understand the relationship between the exterior and interior angles. The Mastering this concept is crucial for the solving more complex geometry problems and proving the various geometric properties.