# The Sum of the Interior Angles of a Triangle - Examples, Exercises and Solutions

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 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.

No.

## examples with solutions for the sum of the interior angles of a triangle

### Exercise #1

Angle A is equal to 30°.
Angle B is equal to 60°.
Angle C is equal to 90°.

Can these angles form a triangle?

### Step-by-Step Solution

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

$30+60+90=180$
The sum of the angles equals 180, so they can form a triangle.

Yes

### Exercise #2

Triangle ADE is similar to isosceles triangle ABC.

Angle A is equal to 50°.

Calculate angle D.

### Step-by-Step Solution

Triangle ABC is isosceles, therefore angle B is equal to angle C. We can calculate them since the sum of the angles of a triangle is 180:

$180-50=130$

$130:2=65$

As the triangles are similar, DE is parallel to BC

Angles B and D are corresponding and, therefore, are equal.

B=D=65

$65$°

### Exercise #3

True or false?

$\alpha+\beta=180$

### Step-by-Step Solution

Given that the angles alpha and beta are on the same straight line and given that they are adjacent angles. Together they are equal to 180 degrees and the statement is true.

True

### Exercise #4

Three angles measure as follows: 60°, 50°, and 70°.

Is it possible that these are angles in a triangle?

### Step-by-Step Solution

Recall that the sum of angles in a triangle equals 180 degrees.

Let's add the three angles to see if their sum equals 180:

$60+50+70=180$

Therefore, it is possible that these are the values of angles in some triangle.

Possible.

### Exercise #5

Find the measure of the angle $\alpha$

### Step-by-Step Solution

Recall that the sum of angles in a triangle equals 180 degrees.

Therefore, we will use the following formula:

$A+B+C=180$

Now let's insert the known data:

$\alpha+50+50=180$

$\alpha+100=180$

We will simplify the expression and keep the appropriate sign:

$\alpha=180-100$

$\alpha=80$

80

## examples with solutions for the sum of the interior angles of a triangle

### Exercise #1

What kind of triangle is shown in the diagram below?

### Step-by-Step Solution

We calculate the sum of the angles of the triangle:

$117+53+21=191$

It seems that the sum of the angles of the triangle is not equal to 180°,

Therefore, the figure can not be a triangle and the drawing is incorrect.

The triangle is incorrect.

### Exercise #2

ABC is an isosceles triangle.

$∢A=4x$

$∢B=2x$

Calculate the value of x.

### Step-by-Step Solution

As we know that triangle ABC is isosceles.

$B=C=2X$

It is known that in a triangle the sum of the angles is 180.

Therefore, we can calculate in the following way:

$2X+2X+4X=180$

$4X+4X=180$

$8X=180$

We divide the two sections by 8:

$\frac{8X}{8}=\frac{180}{8}$

$X=22.5$

22.5

### Exercise #3

Triangle ABC isosceles.

AB = BC

Calculate angle ABC and indicate its type.

### Step-by-Step Solution

Given that it is an isosceles triangle:$AB=BC$

It is possible to argue that:$BAC=ACB=45$

Since the sum of the angles of a triangle is 180, the angle ABC will be equal to:

$180-45-45=90$

Since the angle ABC measures 90 degrees, it is a right triangle.

90°, right angle.

### Exercise #4

It is known that angles A and D are equal to 90 degrees

Angle DEB is equal to 95 degrees

Complete the value of angle GDC based on the data from the figure.

### Step-by-Step Solution

Note that the GDC angle is part of the EDC angle.

Therefore, we can write the following expression:

$GDC+EDG=EDC$

Since we know that angle D equals 90 degrees, we will substitute the values in the formula:

$GDC+40=90$

We will simplify the expression and keep the appropriate sign:

$GDC=90-40$

$GDC=50$

50

### Exercise #5

ABC Right triangle

Since BD is the median

Given AC=10.

Find the length of the side BD.

### Step-by-Step Solution

We calculate BD according to the rule:

In a right triangle, the midpoint of the hypotenuse is equal to half of the hypotenuse.

That is:

BD is equal to half of AC:

Given that: $AC=10$

$BD=10:2=5$