Exterior Angle Theorem

Complete Guide to Alternate Exterior Angles Theorem With Triangle Examples

exterior-angle-theoremWhat is Exterior Angle Theorem?

Contents

Definition: Exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of two non-adjacent interior angles (opposite interior angles).

An exterior angle of a triangle is formed by the extension of any one side of the triangle. The exterior angle is not just outside the triangle but it is also adjacent to an interior angle.

Exterior angle theorem also states that the measure of an exterior angle of a triangle is greater than either of the two opposite interior angles (remote interior angles).


Alternate Exterior Angles Theorem

Characteristics

  • Exterior angle is always equal to the sum of the opposite interior angles.
  • Exterior angle is always greater than the either of the two remote interior angles.
  • Exterior angle is always supplementary to its adjacent interior angle.

Uses

Exterior angle theorem could be used to find the measures of the unknown interior and exterior angles of a triangle.

Importance

Exterior angle theorem is one of the important theorems of the triangle. With the help of exterior angle theorem, unknown interior and exterior angles of a triangle can be found easily.


Triangle Exterior Angle Theorem Formula

exterior-angle-theorem-example

As shown in the figure above, interior angles of the triangle are angle 1, angle 2 and angle 3.

Angle 4 is the exterior angle adjacent to the angle 3.

Angle 1 and angle 2 are the opposite interior angles (remote interior angles) to the exterior angle 4. External theorem states that the measure of an exterior angle of a triangle is equal to the sum of two remote interior angles (opposite interior angles).

m∠1 + m∠2 = m∠4

Exterior angle theorem states that the measure of an exterior angle of a triangle is greater than either of the two opposite interior angles.

m∠4 > m∠1

m∠4 > m∠2


What are exterior angles of a triangle?

The exterior angles of a triangle are the angles that form an adjacent pair with the interior angles by extending the sides of the triangle.

Example

In the triangle given below, external angles and internal angles are shown.

alternate-exterior-angle-theorem


Exterior Angle Theorem Examples

How to find exterior angles

Example 1

Triangle ABC, m∠B = 45°, and m∠C = 75°. Find the exterior angles.

Solution:

Measure of exterior angle adjacent to angle A = m∠B + m∠C = 45° + 75° = 120°.

To find the measure of other exterior angles

First find the unknown interior angle.

We know that the sum of the interior angles of a triangle = 180°.

m∠A + m∠B + m∠C = 180°

m∠A + 45° + 75° = 180°

m∠A = 60°

Measure of exterior angle adjacent to angle B = m∠A + m∠C = 60° + 75° = 135°.

Measure of exterior angle adjacent to angle C = m∠A + m∠B = 60° + 45° = 105°.


Example 2

In Triangle ABC, an exterior angle at D is represented by 5x + 11. If the two non-adjacent interior angles are represented by 2x + 8, and 4x – 17, find the value of x.

Solution:

The Exterior Angle theorem states that measure of an exterior angle of a triangle is equal to the sum of two non-adjacent interior angles.

Therefore,

5x + 11 = (2x + 8) + (4x – 17)

5x + 11 = 6x – 9

x = 20


Example 3

Find the measure of an exterior angle at the base of an isosceles triangle whose vertex angle measures 35°.

triangle-exterior-angle-theorem

Solution:

As we know that the two sides of an isosceles triangle are equal. Angles opposite to the equal angles are also equal. The two base angles of an isosceles triangle are equal, so we can assume each as x.

x + x + 35 = 180 (Sum of the interior angles of a triangle is equal to 180°).

2x + 35 = 180

2x = 180 – 35

2x = 145

x = 72.5

So, the exterior angle is equal to the sum of the two non-adjacent interior angles.

Therefore,

? = 72.5° + 35°

? = 107.5°


Example 4

Find x in the triangle given below and hence find m∠ABD.

triangle-exterior-angle-theorem-example

Solution:

∠C and ∠D are non-adjacent interior angles for the exterior angle ABD. As the exterior angle theorem states that the exterior angle is equal to the sum of the two non-adjacent interior angles.

∠ABD = ∠C + ∠D

20x = 7x + 5 + 60

20x – 7x = 65

13x = 65

x = 5

m∠ABD = 20x

m∠ABD = 20(5)

m∠ABD = 100°