What theorem states that the measure of an interior angle?

Jana Russick

Image credit: Desmos

Before we cover the exterior angle theorem, let's review a few definitions.

• Adjacent angles: angles that share a side and a vertex (ex., BCA and DCA)
• Supplementary angles: two angles that add to 180°
• Interior angles: the angles inside a triangle
• Exterior angles: angles formed between a side of a shape and a line that extends from the next side

We'll use the above triangle to demonstrate the exterior angle theorem's principles:

1. An exterior angle should equal the sum of the remote interior angles of a triangle. In the triangle above, the exterior angle of the triangle, angle ACD, will equal the sum of the measures of interior angles BAC and ABC.
2. An exterior angle and its adjacent interior angle are supplementary angles, so they add to 180°. Above, BCA plus ACD add to 180°.

Breaking Down the Exterior Angle Theorem

Let's look at how the exterior angle theorem works. First, let’s review the angle sum theorem, which states that the interior angles of a triangle equal 180°.

Image credit: Desmos

In the above triangle ECD, the exterior angle of DEF and its adjacent interior angle CED are linear pairs. That means together, they form a straight line and equal 180°.

Because these two adjacent angles add to 180° and the interior measures of the angles of a triangle also equal 180°, the sum of the remote interior angles ECD and CDE must equal the measure of exterior angle DEF.

Next, we'll use this knowledge to find angle measurements.

Applying the Exterior Angle Theorem

Let's use the exterior angle theorem in the triangle below:

Image credit: Desmos

Since we know that the angle EST = 125° and the adjacent interior angle TSU is its supplementary angle, let's solve for the measure of this interior angle:

Now let's use the second part of the exterior angle theorem: The exterior angle equals the sum of the remote interior angles. We'll follow this logic and find the remote interior angle TUS by subtracting STU from EST:

Understanding Exterior and Interior Angles

The exterior angle theorem states that:

• The measure of an exterior angle of a triangle is supplementary to its adjacent interior angle.
• The sum of the remote interior angles must equal the measure of the exterior angle of the triangle.

This theorem can help you solve for missing angles and understand the relationship between exterior and interior angles within a triangle.

More Math Homework Help:

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

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.

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°.

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.

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°