# Exterior angle of a triangle

🏆Practice parts 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.
The exterior angle is equal to the sum of the two interior angles of the triangle that are not adjacent to it.

It is defined as follows:

$α=∢A+∢B$

## Test yourself on parts of a triangle!

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

## Exterior angle of a triangle

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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## What does the continuation of the side mean?

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?

### Examples of Exterior Angles

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, $∢A+∢B$

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 $70$.

$180-110=70$

Now, let's remember that the sum of the interior angles of a triangle is $180$

and we can find $∢B$

$∢B=180-90-70$
$∢B=20$

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

### examples.example_title

Which of the following is the height in triangle ABC?

### examples.explanation_title

Let's remember the definition of height:

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

Therefore, the one that forms a 90-degree angle is side AB with side BC

AB

### examples.example_title

Given the isosceles triangle ABC,

The side AD is the height in the triangle ABC

and inside it, EF is drawn:

AF=5 AB=17

What is the perimeter of the trapezoid EFBC?

### examples.explanation_title

To find the perimeter of the trapezoid, all its sides must be added:

We will focus on finding the bases.

To find GF we use the Pythagorean theorem: $A^2+B^2=C^2$in the triangle AFG

We replace

$3^2+GF^2=5^2$

We isolate GF and solve:

$9+GF^2=25$

$GF^2=25-9=16$

$GF=4$

We perform the same process with the side DB of the triangle ABD:

$8^2+DB^2=17^2$

$64+DB^2=289$

$DB^2=289-64=225$

$DB=15$

We start by finding FB:

$FB=AB-AF=17-5=12$

Now we reveal EF and CB:

$GF=GE=4$

$DB=DC=15$

This is because in an isosceles triangle, the height divides the base into two equal parts so:

$EF=GF\times2=4\times2=8$

$CB=DB\times2=15\times2=30$

All that's left is to calculate:

$30+8+12\times2=30+8+24=62$

62