ABC is a triangle. What is the median of the triangle? A A A B B B C C C E E E F F F D D D

There is no median in the figure.

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!

ABC is a triangle.

What is the median of the triangle?

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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

Fill in the blanks:

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

Question 2

Given two triangles, Is EB a side of one of the triangles?

Question 3

In an isosceles triangle, the angle between ? and ? is the "base angle".

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

In an isosceles triangle, the third side is called?

Question 2

Look at the triangle ABC below.

$AD=\frac{1}{2}AB$

$BE=\frac{1}{2}EC$

What is the median in the triangle?

Question 3

Look at the triangle ABC below.

Which of the following lines is the median of the triangle?

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 triangle ABC below.

Which of the line segments is the median?

Question 2

Look at the triangles in the figure.

Which line is the median of triangle ABC?

Question 3

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

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

Exercise #1

Determine the type of angle given.

Video Solution

Step-by-Step Solution

To solve this problem, we'll examine the image presented for the angle type:

Step 1: Identify the angle based on the visual input provided in the graphical representation.
Step 2: Classify it using the standard angle types: acute, obtuse, or straight based on their definitions.
Step 3: Select the appropriate choice based on this classification.

Now, let's apply these steps:

Step 1: Analyzing the provided diagram, observe that there is an angle formed among the segments.

Step 2: The angle is depicted with a measure that appears greater than a right angle (greater than $90^\circ$ ). It is wider than an acute angle.

Step 3: Given the definition of an obtuse angle (greater than $90^\circ$ but less than $180^\circ$ ), the graphic clearly shows an obtuse angle.

Therefore, the solution to the problem is Obtuse.

Answer

Obtuse

Exercise #2

Indicates which angle is greater

Video Solution

Step-by-Step Solution

Note that in drawing B, the two lines form a right angle, which is an angle of 90 degrees:

While the angle in drawing A is greater than 90 degrees:

Therefore, the angle in drawing A is larger.

Answer

Exercise #3

Indicates which angle is greater

Video Solution

Step-by-Step Solution

In drawing A, we can see that the angle is an obtuse angle, meaning it is larger than 90 degrees:

While in drawing B, the angle is a right angle, meaning it equals 90 degrees:

Therefore, the larger angle appears in drawing A.

Answer

Exercise #4

Which angle is greater?

Video Solution

Step-by-Step Solution

The angle in diagram (a) is more acute, meaning it is smaller:

Conversely, the angle in diagram (b) is more obtuse, making it larger.

Answer

Exercise #5

Indicates which angle is greater

Video Solution

Step-by-Step Solution

Answer B is correct because the more closed the angle is, the more acute it is (less than 90 degrees), meaning it's smaller.

The more open the angle is, the more obtuse it is (greater than 90 degrees), meaning it's larger.

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