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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Can a triangle have a right angle?

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

Question 2

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

Question 3

Can a plane angle be found in a triangle?

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

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

Question 2

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

Question 3

Is the straight line in the figure the height 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

Can a triangle have a right angle?

Video Solution

Step-by-Step Solution

To determine if a triangle can have a right angle, consider the following explanation:

Definition of a Right Angle: An angle is classified as a right angle if it measures exactly $90^\circ$ .
Definition of a Right Triangle: A right triangle is a type of triangle that contains exactly one right angle.
According to the definition, a right triangle specifically includes a right angle. This is a well-established classification of triangles in geometry.

Thus, a triangle can indeed have a right angle and is referred to as a right triangle.

Therefore, the solution to the problem is Yes.

Answer

Yes

Exercise #2

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

Video Solution

Step-by-Step Solution

The triangle's altitude is a line drawn from a vertex perpendicular to the opposite side. The vertical line in the diagram extends from the triangle's top vertex straight down to its base. By definition of altitude, this line is the height if it forms a right angle with the base.

To solve this problem, we'll verify that the line in question satisfies the altitude condition:

Step 1: Identify the triangle's vertices and base. From the diagram, the base appears horizontal, and the vertex lies directly above it.
Step 2: Check the nature of the line. The line is vertical when the base is horizontal, indicating perpendicularity.
Conclusion: The vertical line forms right angles with the base, thus acting as the altitude or height.

Therefore, the straight line depicted is indeed the height of the triangle. The answer is Yes.

Answer

Yes

Exercise #3

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

Video Solution

Step-by-Step Solution

To determine if the straight line is the height of the triangle, we'll analyze its role within the triangle:

Step 1: Observe the triangle and the given line. The triangle seems to be made of three sides and a vertical line within it.
Step 2: Recall that the height of a triangle, in geometry, is defined as a perpendicular dropped from a vertex to the opposite side.
Step 3: Examine the positioning of the line: The vertical line starts at one vertex of the triangle and intersects the base, appearing to be perpendicular.
Step 4: Verify perpendicularity: Given that the line is shown as a clear vertical (and a small perpendicular indicator suggests perpendicularity), we accept this line as the height.
Step 5: Conclude with verification that the line is effectively meeting the definition of height for the triangle in the diagram.

Therefore, the vertical line in the figure is indeed the height of the triangle.

Yes

Answer

Yes

Exercise #4

Can a plane angle be found in a triangle?

Video Solution

Step-by-Step Solution

To determine whether a plane angle can be found in a triangle, we need to understand what a plane angle is and compare it to the angles within a triangle.

A plane angle is an angle formed by two lines lying in the same plane.
In the context of geometry, angles found within a triangle are the interior angles, which are the angles between the sides of the triangle.
Although the angles in a triangle are indeed contained within a plane (since a triangle itself is a planar figure), when referencing "plane angles" in geometry, we usually consider angles related to different geometric configurations beyond those specifically internal to defined planar shapes like a triangle.
The term "plane angle" typically refers to the measurement of an angle in radians or degrees within a plane, but this doesn't specifically pertain to angles of a triangle.

Therefore, based on the context and usual geometric conventions, the concept of a "plane angle" is not typically used to describe the angles found within a triangle. Thus, a plane angle as defined generally in geometry is not found specifically within a triangle.

Therefore, the correct answer to the question is $\text{No}$ .

Answer

Exercise #5

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

Video Solution

Step-by-Step Solution

In the given problem, we have a triangle depicted with a specific line drawn inside it. The question asks if this line represents the height of the triangle. To resolve this question, we need to discern whether the line is perpendicular to one of the sides of the triangle when extended, as only a line that is perpendicular from a vertex to its opposite side can be considered the height.

The line in question is shown intersecting one of the sides within the triangle but does not form a perpendicular angle with any side shown or the ground (as is required for it to be the height of the triangle). A proper height would typically intersect perpendicularly either at or along the extended line of the opposite side from a vertex.

Therefore, based on the visual clues provided and the typical geometric definition of a height (or altitude) in a triangle, this specific line does not fit the criteria for being a height.

Thus, we conclude that the line depicted is not the height of the triangle. The correct answer is No.

Answer

Check your understanding

Question 1

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

Question 2

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

Question 3

According to figure BC=CB?

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Exterior angles of a triangle