The Sum of the Interior Angles of a Triangle

The sum of the interior angles of a triangle is $180º$ . If we add the three angles of any triangle we choose, the result will always be $180º$ . This means that if we know the values of two angles of a triangle we can always calculate, with ease, the value of the third one: first we add the two angles we know and then we subtract from $180º$ The result of this subtraction will give us the value of the third angle of the triangle.

For example, given a triangle with two known interior angles of $45º$ and $60º$ degrees, we are asked to discover the measure of the third angle. First we add $45º$ plus $60º$ resulting in $105º$ degrees. Now we subtract $105º$ from $180º$ , yielding $75º$ degrees. In other words, the third angle of the triangle equals $75º$ degrees.

The above property is also called the triangle sum theorem, and can help us to solve problems involving the interior angles of a triangle, regardless of whether it is equilateral, isosceles or scalene.

Examples of different types of triangles and the sum of the interior angles in each

Isosceles triangle:

Equilateral triangle:

Scalene triangle:

Detailed explanation

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Test yourself on parts of a triangle!

Fill in the blanks:

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

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Questions on the subject

What does the triangle sum theorem tell us?

The theorem tells us that the sum of the interior angles of any triangle is equal to 180°.

How do we find the third interior angle of a triangle, knowing the other two?

By applying the theorem, we subtract the sum of the two given angles from 180°.

How much must the interior angles of a triangle add up to?

180°.

Exercises for addition of the interior angles of a triangle:

Exercise 1

Task:

Given three angles:

Angle $A$ is equal to $30°$

Angle $B$ is equal to $60°$

Angle $C$ is equal to $90°$

Can these angles form a triangle?

Solution

It is known that the sum of the angles of the triangles must be equal to $180°$

Let's add the value of the angles and see if together they are equal to $180°$

$A+B+C=30+60+90=180$

Answer

Yes

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Test your knowledge

Question 1

The sum of the adjacent angles is 180

Question 2

Is DE side in one of the triangles?

Question 3

Look at the two triangles below. Iis EB a side of one of the triangles?

Exercise 2

Task:

Given three angles:

Angle $A$ is equal to $60°$

Angle $B$ is equal to $60°$

Angle $C$ is equal to $60°$

Can these angles form a triangle?

Solution

It is known that the sum of the angles of the triangles must be equal to $180°$

Let's add the value of the angles and see if together they are equal to $180°$

$A+B+C=60+60+60=180$

Answer

Yes

Exercise 3

Task:

Given three angles:

Angle A is equal to $90°$

Angle B is equal to $115°$

Angle C is equal to $35°$

Can these angles form a triangle?

Solution

We know that the sum of the angles of the triangle must be equal to $180°$

We add the total of the angles to see if together they are equal to $180°$

$A+B+C=90+115+35=240$

We observe that the sum of the three angles are equal to $240°$ , that is to say that they cannot form a triangle.

Answer

Do you know what the answer is?

Question 1

Look at the two triangles below.

Is AD a side of one of the triangles?

Question 2

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

Question 3

Look at the two triangles below.

Is AB a side of one of the triangles?

Exercise 4

Assignment:

Given the parallel lines.

Find the angle $\alpha$

Solution

The angle beta is equal to $90°$ . The adjacent angle is also equal to $90°$ since the sum is equal to $180°$ degrees. The adjacent angle gamma $120°$ and their sum is equal to $180°$ , therefore, gamma is equal to $60°$ degrees.

$\alpha+\gamma+\delta=180°$

$\alpha+60°+90°=180°$

$\alpha+150°=180°$

$\alpha=180°-150°$

$\alpha=30°$

Answer

$30°$

Exercise 5

$CE$ is parallel to $AD$

What is the value of $X$ if it is given that $ABC$ is isosceles, such that $AB=BC$

Solution

Angles $\sphericalangle UCH$ and angle $\sphericalangle ACE$ are opposite angles.

$\text{AC}E=\text{ICH}=2X$

$\sphericalangle DAC$ and angle $\sphericalangle\text{AC}E$ are collateral angles.

$2x+\text{DAC}=180$

$\text{DAC}=180-2x$

$\sphericalangle FGA$ and angle $\sphericalangle DAB$ are opposite angles.

$\text{FGA}=\text{DAB}=x-10$

$\text{BAC}=\text{DAC}-\text{DAB}=$

$180-2x-(x-10)=$

$190-3x$

The sum of the angles in the triangle is $180$

$\text{ACB}+\text{CAB}+B=180$

$\text{ACB}=180-(190-3x)-(3x-30)=20$

$\text{ACB}=\text{BAC}$

$20=190-3x$

$x=56.67$

Answer

$56.67$

Check your understanding

Question 1

Look at the two triangles below.

Is BC a side of one of the triangles?

Question 2

Look at the two triangles below.

Is CB a side of one of the triangles?

Question 3

Look at the two triangles below.

Is DF a side in one of the triangles?

Examples with solutions for The Sum of the Interior Angles 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 image it is clear that the above theorem is true for the line AE. AE not only connects the A vertex with the opposite side. It also crosses BC forming a 90-degree angle. Undoubtedly making AE the altitude.

Answer

Exercise #2

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

Video Solution

Step-by-Step Solution

Every triangle has 3 sides. First let's go over the triangle on the left side:

Its sides are: AB, BC, and CA.

This means that in this triangle, side EC does not exist.

Let's then look at the triangle on the right side:

Its sides are: ED, EF, and FD.

This means that in this triangle, side EC also does not exist.

Therefore, EC is not a side in either of the triangles.

Answer

Exercise #3

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

Exercise #4

In a right triangle, the sum of the two non-right angles is...?

Video Solution

Step-by-Step Solution

In a right-angled triangle, there is one angle that equals 90 degrees, and the other two angles sum up to 180 degrees (sum of angles in a triangle)

Therefore, the sum of the two non-right angles is 90 degrees

$90+90=180$

Answer

90 degrees

Exercise #5

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

Start practice

The Sum of the Interior Angles of a Triangle

Examples of different types of triangles and the sum of the interior angles in each

Test yourself on parts of a triangle!

Questions on the subject

Exercises for addition of the interior angles of a triangle:

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Examples with solutions for The Sum of the Interior Angles of a Triangle

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

More Questions

Sum and Difference of Angles

Parts of a Triangle