The exterior angle of a triangle is the one that is found between the original side and the extension of the side. The exterior angle is equal to the sum of the two interior angles of the triangle that are not adjacent to it.

Until today we have dealt with internal angles, perhaps also with adjacent angles, but we have not talked about external angles. Don't worry, the topic of the exterior angle of a triangle is very easy to understand and its property can be very useful for solving exercises more quickly. Shall we start?

What is the exterior angle of a triangle?

The exterior angle of a triangle is the one that is found between the original side and the extension of the side.

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Test your knowledge

Question 1

Is the straight line in the figure the height of the triangle?

Imagine someone draws a triangle and falls asleep as they are finishing it. Without realizing it, they continue drawing one side a little more... and Bam! An exterior angle is created. The exterior angle is outside of the triangle and is found between the original side and the side they continued drawing while asleep (the continuation of the side).

Let's look at an example

Observe: An exterior angle is the one that is found between an original side of the triangle and the extension of the side and not between two extensions.

Note: Whenever the angle is outside the triangle and is found between an original side of the triangle and the extension of another side of the triangle, it will be considered an exterior angle of the triangle.

Do you know what the answer is?

Question 1

Is the straight line in the figure the height of the triangle?

Great! Now that we have understood what an exterior angle is and that we can recognize it from a distance, we can move on to the property of the exterior angle of a triangle. Property of the exterior angle of a triangle The exterior angle is equal to the sum of the two interior angles of the triangle that are not adjacent to it.

Given that: $∢A=80$ $∢B=20$

How much does the exterior angle measure? Solution: Let's denote the exterior angle with$α$:

According to the property of the exterior angle of the triangle, the exterior angle $α$ must be equal to the sum of the two interior angles of the triangle that are not adjacent to it. That is, $∢A+∢B$

We already have these angles. Therefore, all we have to do is add them up and find out the exterior angle: $α=80+20$ $α=100$

Look, we could have found the value of the exterior angle in another way! We know that the sum of the interior angles of a triangle is $180$. Therefore, $∢ACB=180-20-80$

$∢ACB=80$

$∢ABC$ is the angle adjacent to $α$, the exterior angle of the triangle that we need to find out. We also know that the sum of the adjacent angles is $180$. Therefore we can determine that: $∢80+α=80$ $α=100$

Look, In certain cases you will not be explicitly asked for the value of the exterior angle. They might ask you, for example, about some interior angle of the triangle that you could figure out through the exterior angle.

Let's look at an example

Given the following triangle:

Data: $∢A=90$ $α=110$

Find the value of $∢B$

Solution:

We can solve the problem in two ways:

The first is based on the Exterior Angle Theorem of a triangle and understand that $α$ is an exterior angle of the triangle and is equal to the sum of the two interior angles that are not adjacent to it. That is, $∢A+∢B$

Then, the equation would be: $110=90+∢B$ $∢B=20$

The second way to solve the problem is to remember that the sum of the adjacent angles equals $180$, then $∢ACB$ is equal to $70$.

Notice that we have arrived at the same result, but solving through the property of the exterior angle of a triangle has been faster to reach it.

Useful Information: The sum of the three exterior angles of a triangle equals $360$ degrees.

In conclusion, it is important and really worth knowing the property of the exterior angle of a triangle to solve problems easily and quickly, although in several cases you will be able to manage without this magnificent theorem.

Examples and exercises with solutions of an exterior angle of a triangle

Exercise #1

Given the following triangle:

Write down the height of the triangle ABC.

Video Solution

Step-by-Step Solution

An altitude in a triangle is the segment that connects the vertex and the opposite side, in such a way that the segment forms a 90-degree angle with the side.

If we look at the drawing, we can notice that the previous theorem is true for the line AE that crosses BC and forms a 90-degree angle, comes out of vertex A and therefore is the altitude of the triangle.

Answer

AE

Exercise #2

ABC is an isosceles triangle.

AD is the median.

What is the size of angle $∢\text{ADC}$?

Video Solution

Step-by-Step Solution

In an isosceles triangle, the median to the base is also the height to the base.

That is, side AD forms a 90° angle with side BC.

That is, two right triangles are created.

Therefore, angle ADC is equal to 90 degrees.

Answer

90

Exercise #3

Which of the following is the height in triangle ABC?

Video Solution

Step-by-Step Solution

Let's remember the definition of height of a triangle:

A height is a straight line that descends from the vertex of a triangle and forms a 90-degree angle with the opposite side.

The sides that form a 90-degree angle are sides AB and BC. Therefore, the height is AB.

Answer

AB

Exercise #4

Angle A equals 56°. Angle B equals 89°. Angle C equals 17°.

Can these angles make a triangle?

Video Solution

Step-by-Step Solution

We add the three angles to see if they are equal to 180 degrees:

$56+89+17=162$

The sum of the given angles is not equal to 180, so they cannot form a triangle.

Answer

No.

Exercise #5

Angle A equals 90°. Angle B equals 115°. Angle C equals 35°.

Can these angles form a triangle?

Video Solution

Step-by-Step Solution

We add the three angles to see if they are equal to 180 degrees:

$90+115+35=240$ The sum of the given angles is not equal to 180, so they cannot form a triangle.

Answer

No.

Check your understanding

Question 1

Is the straight line in the figure the height of the triangle?