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 180º . If we add the three angles of any triangle we choose, the result will always be 180º 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 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 45º and 60º 60º degrees, we are asked to discover the measure of the third angle. First we add 45º 45º plus 60º 60º resulting in 105º 105º degrees. Now we subtract 105º 105º from 180º 180º , yielding 75º 75º degrees. In other words, the third angle of the triangle equals 75º 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

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Test yourself on parts of a triangle!

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Can a triangle have a right 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 A is equal to 30° 30°

Angle B B is equal to 60° 60°

Angle C C is equal to 90° 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° 180°

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

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

Answer

Yes


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Exercise 2

Task:

Given three angles:

Angle A A is equal to 60° 60°

Angle B B is equal to 60° 60°

Angle C C is equal to 60° 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° 180°

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

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

Answer

Yes


Exercise 3

Task:

Given three angles:

Angle A is equal to 90° 90°

Angle B is equal to 115° 115°

Angle C is equal to 35° 35°

Can these angles form a triangle?

Solution

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

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

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

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

Answer

No


Do you know what the answer is?

Exercise 4

Assignment:

Exercise 3 Assignment Given the following parallel lines

Given the parallel lines.

Find the angle α \alpha

Solution

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

α+γ+δ=180° \alpha+\gamma+\delta=180°

α+60°+90°=180° \alpha+60°+90°=180°

α+150°=180° \alpha+150°=180°

α=180°150° \alpha=180°-150°

α=30° \alpha=30°

Answer

30° 30°


Exercise 5

CE CE is parallel to AD AD

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

Exercise 4 CE is parallel to AD

Solution

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

are opposite at the vertex

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

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

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

DAC=1802x \text{DAC}=180-2x

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

FGA=DAB=x10 \text{FGA}=\text{DAB}=x-10

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

1802x(x10)= 180-2x-(x-10)=

1903x 190-3x

The sum of the angles in the triangle is 180 180

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

ACB=180(1903x)(3x30)=20 \text{ACB}=180-(190-3x)-(3x-30)=20

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

20=1903x 20=190-3x

x=56.67 x=56.67

Answer

56.67 56.67


Check your understanding

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.

AAABBBCCCEEEDDD

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 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.

Answer

AE

Exercise #2

ABC is an isosceles triangle.

AD is the median.

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

AAABBBCCCDDD

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

90

Exercise #3

Which of the following is the height in triangle ABC?

AAABBBCCCDDD

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

AB

Exercise #4

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

Can these angles make a triangle?

Video Solution

Step-by-Step Solution

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

56+89+17=162 56+89+17=162

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

Answer

No.

Exercise #5

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

Can these angles form a triangle?

Video Solution

Step-by-Step Solution

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

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

Answer

No.

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