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!

True or false:

DE not a side in any of the triangles.

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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

Is DE side in one of the triangles?

Question 2

What is the size of the unlabelled angle?

Question 3

What is the size of the missing 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

What is the size of the missing angle?

Question 2

The triangle ABC is shown below.

To which side(s) are the median and the altitude drawn?

Question 3

The triangle ABC is shown below.

Which line segment is the median?

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

True or false:

DE not a side in any of the triangles.

Video Solution

Step-by-Step Solution

To solve the problem of determining whether DE is not a side in any of the triangles, we will methodically identify the triangles present in the diagram and examine their sides:

Identify triangles in the diagram. The diagram presented forms a right-angled triangle ABC with additional lines forming smaller triangles within.
Triangles formed: Triangle ABC (major triangle), Triangle ABD, Triangle BEC, and Triangle DBE.
Let's examine the sides of these triangles:

Triangle ABC has sides AB, BC, and CA.
Triangle ABD has sides AB, BD, and DA.
Triangle BEC has sides BE, EC, and CB.
Triangle DBE has sides DB, BE, and ED.

Notice that while point D is used, the segment DE is only part of line BE and isn't listed as a direct side of any triangle.

Therefore, the claim that DE is not a side in any of the triangles is indeed correct.

Hence, the answer is True.

Answer

True

Exercise #2

Is DE side in one of the triangles?

Video Solution

Step-by-Step Solution

Since line segment DE does not correspond to a full side of any of the triangles present within the given geometry, we conclude that the statement “DE is a side in one of the triangles” is Not true.

Answer

Not true

Exercise #3

What is the size of the missing angle?

Video Solution

Step-by-Step Solution

To find the size of the missing angle, we will use the property that the sum of angles on a straight line is $180^\circ$ . Given that one angle is $80^\circ$ , we can calculate the missing angle using the following steps:

Step 1: Recognize that the given angle $\alpha = 80^\circ$ and the missing angle $\beta$ form a straight line.
Step 2: Use the angle sum property for a straight line: $\alpha + \beta = 180^\circ$
Step 3: Substitute the known value: $80^\circ + \beta = 180^\circ$
Step 4: Solve for the missing angle $\beta$ : $\beta = 180^\circ - 80^\circ = 100^\circ$

Therefore, the size of the missing angle is $100^\circ$ .

Answer

100°

Exercise #4

The triangle ABC is shown below.

To which side(s) are the median and the altitude drawn?

Step-by-Step Solution

To solve the problem of identifying to which side of triangle $ABC$ the median and the altitude are drawn, let's analyze the diagram given for triangle $ABC$ .

We acknowledge that a median is a line segment drawn from a vertex to the midpoint of the opposite side. An altitude is a line segment drawn from a vertex perpendicular to the opposite side.
Upon reviewing the diagram of triangle $ABC$ , line segment $AD$ is a reference term. It appears to meet point $C$ in the middle, suggesting it's a median, but it also forms right angles suggesting it is an altitude.
Given the placement and orientation of $AD$ , it is perpendicular to line $BC$ (the opposite base for the median from $A$ ). Therefore, this line is both the median and the altitude to side $BC$ .

Thus, the side to which both the median and the altitude are drawn is BC.

Therefore, the correct answer to the problem is the side $BC$ , corresponding with choice $\text{Option 2: BC}$ .

Answer

Exercise #5

The triangle ABC is shown below.

Which line segment is the median?

Video Solution

Step-by-Step Solution

To solve this problem, we need to identify the median in triangle ABC:

Step 1: Recall the definition of a median. A median is a line segment drawn from a vertex to the midpoint of the opposite side.
Step 2: Begin by evaluating each line segment based on the definition.
Step 3: Identify points on triangle ABC:

AD is from A to a point on BC.
BE is from B to a point on AC.
FC is from F to a point on AB.

Step 4: Determine if these points (D, E, F) are midpoints:

Since BE connects B to E, and E is indicated to be the midpoint of segment AC (as shown), BE is the median.
AD and FC, by visual inspection, do not connect to midpoints on BC or AB respectively.

Therefore, the line segment that represents the median is $BE$ .

Thus, the correct answer is: BE

Answer

Check your understanding

Question 1

Look at triangle ABC below.

What is the median of the triangle and to which side is it drawn?

Question 2

Look at triangle ABC below.

Which is the median?

Question 3

Look at the triangle ABC below.

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

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

What is the median in the triangle?

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Exterior angles of a triangle