# The Sum of the Interior Angles of a Triangle

🏆Practice parts 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.

## Test yourself on parts of a triangle!

Can a triangle have a right angle?

## 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

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 $180°$

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

Yes

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### Exercise 2

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 $180°$

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

Yes

### Exercise 3

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.

No

Do you know what the answer is?

### 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.

$\alpha+\gamma+\delta=180°$

$\alpha+60°+90°=180°$

$\alpha+150°=180°$

$\alpha=180°-150°$

$\alpha=30°$

$30°$

### 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 $180$

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

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

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

$20=190-3x$

$x=56.67$

$56.67$

## 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.

### 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 drawing, we can notice that the previous theorem is true for the line AE that crosses BC and forms a 90-degree angle, comes out of vertex A and therefore is the altitude of the triangle.

AE

### Exercise #2

ABC is an isosceles triangle.

What is the size of angle $∢\text{ADC}$?

### 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.

90

### Exercise #3

Which of the following is the height in triangle ABC?

### 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.

AB

### Exercise #4

Angle A equals 56°.
Angle B equals 89°.
Angle C equals 17°.

Can these angles make a triangle?

### Step-by-Step Solution

We add the three angles to see if they are equal to 180 degrees:

$56+89+17=162$

The sum of the given angles is not equal to 180, so they cannot form a triangle.

No.

### Exercise #5

Angle A equals 90°.
Angle B equals 115°.
Angle C equals 35°.

Can these angles form a triangle?

### Step-by-Step Solution

We add the three angles to see if they are equal to 180 degrees:

$90+115+35=240$
The sum of the given angles is not equal to 180, so they cannot form a triangle.