The exterior angles of a triangle are formed when we extend the sides of the triangle. The sum of exterior angles of a triangle is always equal to 360°. Therefore, depending on the type of triangle, we can apply different methods to find the measure of each angle.

Here, we will learn how to determine the measure of the exterior angles of different types of triangles. In addition, we will look at some practice examples.

## Sum of exterior angles of a triangle

The sum of the exterior angles of any polygon is always equal to 360°. This is because, if we join the exterior angles, we will form a complete circle, which represents 360°. We can see this in the following diagram.

Exterior angles are formed by extending the sides of the triangle. Depending on the type of triangle, the measurements of each exterior angle will change, but the sum will always remain the same.

## Calculate the exterior angles of an equilateral triangle

An equilateral triangle is a triangle that has all its sides with the same length and all its interior angles with the same measure. This means that all of its exterior angles also have the same measure.

Since the sum total of the exterior angles equals 360°, we can divide by 3 to get the measure of each exterior angle in an equilateral triangle. Therefore, we have:

360°÷3 = 120°

Each external angle measures 120°.

## Calculate the exterior angles of an isosceles triangle

An isosceles triangle is a triangle that has two sides of the same length and the third side of a different length. Similarly, these triangles have two interior angles with the same measure and the third angle with a different measure. This means that two exterior angles will have the same measure and one exterior angle will have a different measure.

To determine the measures of the exterior angles, we need the measure of at least one exterior or interior angle. We use the facts that the sum of the exterior angles is equal to 360° and that the sum of an interior angle and its corresponding exterior angle is equal to 180°.

### EXAMPLE 1

Determine the measures of the missing exterior angles.

**Solution:** The angles that are marked with a double line are equal, so we have *b*=130°. To find the measure of angle *a*, we have to add the known angle measures and subtract from 360°. Therefore, we have:

130°+130° = 260°

⇒ 360°-260° = 100°

Angle *a* measures 100°.

### EXAMPLE 2

What are the measures of all the exterior angles of the isosceles triangle?

**Solution:** In this case, we have the measure of an internal angle. Therefore, we can find the measure of its corresponding exterior angle by subtracting the angle from 180°:

180°-70° = 110°

The measure of angle *a* is 110°. Now, we subtract this angle from 360° to find the measures of the other two exterior angles:

360°-110° = 250°

This corresponds to the sum of both angles. Since both angles are equal, we divide the sum by 2 and we have:

250°÷2 = 125°

Angles *b* and *c* measure 125°.

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## Calculate the exterior angles of a scalene triangle

Scalene triangles have all their sides with different lengths and all of their interior angles with different measures. This means that all of its exterior angles also have different measures.

Therefore, to calculate the measure of an exterior angle, we need to know the measure of two interior angles or two exterior angles.

### EXAMPLE 1

Determine the measure of the missing exterior angle in the scalene triangle below.

**Solution:** We add the measures of the known angles and subtract from 360°. Therefore, we have:

110°+130° = 240°

⇒ 360°-240° = 120°

The missing angle measures 120°.

### EXAMPLE 2

What are the measures of all the exterior angles in the scalene triangle?

**Solution:** In this case, we have two measures of interior angles. We can calculate the measures of their corresponding exterior angles by subtracting them from 180°:

180°-60° = 120°

⇒ *a *= 120°

180°-50° = 130°

⇒ *b* = 130°

Now, we add these angles and subtract them from 360° to get the measure of the third:

120°+130° = 250°

⇒ 360°-250° = 110°

The measure of angle *c* is 110°.

## See also

Interested in learning more about angles of a triangle and other polygons? Take a look at these pages:

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