ABC is a triangle. What is the median of the triangle? A A A B B B C C C E E E F F F D D D

There is no median in the figure.

Exterior angles 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.

Key Properties:

The exterior angle is equal to the sum of the two interior angles of the triangle that are not adjacent to it.
Each exterior angle is supplementary to its adjacent interior angle, meaning their sum is $180^\circ$ .
The sum of all exterior angles of a triangle is always $360^\circ$ , no matter the shape of the triangle.

It is defined as follows:

$α=∢A+∢B$

Detailed explanation

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Test yourself on parts of a triangle!

ABC is a triangle.

What is the median of the triangle?

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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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Test your knowledge

Question 1

Fill in the blanks:

In an isosceles triangle, the angle between two ___ is called the "___ angle".

Question 2

Given two triangles, Is EB a side of one of the triangles?

Question 3

In an isosceles triangle, the angle between ? and ? is the "base angle".

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?

Question 1

In an isosceles triangle, the third side is called?

Question 2

Look at the triangle ABC below.

$AD=\frac{1}{2}AB$

$BE=\frac{1}{2}EC$

What is the median in the triangle?

Question 3

Look at the triangle ABC below.

Which of the following lines is the median of the triangle?

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, $∢A+∢B$

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 $70$ .

$180-110=70$

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

Exercise #1

Determine the type of angle given.

Video Solution

Step-by-Step Solution

To solve this problem, we'll examine the image presented for the angle type:

Step 1: Identify the angle based on the visual input provided in the graphical representation.
Step 2: Classify it using the standard angle types: acute, obtuse, or straight based on their definitions.
Step 3: Select the appropriate choice based on this classification.

Now, let's apply these steps:

Step 1: Analyzing the provided diagram, observe that there is an angle formed among the segments.

Step 2: The angle is depicted with a measure that appears greater than a right angle (greater than $90^\circ$ ). It is wider than an acute angle.

Step 3: Given the definition of an obtuse angle (greater than $90^\circ$ but less than $180^\circ$ ), the graphic clearly shows an obtuse angle.

Therefore, the solution to the problem is Obtuse.

Answer

Obtuse

Exercise #2

Indicates which angle is greater

Video Solution

Step-by-Step Solution

Note that in drawing B, the two lines form a right angle, which is an angle of 90 degrees:

While the angle in drawing A is greater than 90 degrees:

Therefore, the angle in drawing A is larger.

Answer

Exercise #3

Indicates which angle is greater

Video Solution

Step-by-Step Solution

In drawing A, we can see that the angle is an obtuse angle, meaning it is larger than 90 degrees:

While in drawing B, the angle is a right angle, meaning it equals 90 degrees:

Therefore, the larger angle appears in drawing A.

Answer

Exercise #4

Which angle is greater?

Video Solution

Step-by-Step Solution

The angle in diagram (a) is more acute, meaning it is smaller:

Conversely, the angle in diagram (b) is more obtuse, making it larger.

Answer

Exercise #5

Indicates which angle is greater

Video Solution

Step-by-Step Solution

Answer B is correct because the more closed the angle is, the more acute it is (less than 90 degrees), meaning it's smaller.

The more open the angle is, the more obtuse it is (greater than 90 degrees), meaning it's larger.

Answer

Check your understanding

Question 1

Look at the triangle ABC below.

Which of the line segments is the median?

Question 2

Look at the triangles in the figure.

Which line is the median of triangle ABC?

Question 3

Look at the two triangles below. Is DE a side of one of the triangles?

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Exterior angles of a triangle