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
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
ABC is a triangle.
What is the median of the triangle?
Incorrect
Correct Answer:
EC
Question 2
AB is a side in triangle ADB
Incorrect
Correct Answer:
True
Question 3
According to figure BC=CB?
Incorrect
Correct Answer:
True
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
No
Do you know what the answer is?
Question 1
AD is the median in triangle ABC.
BD = 4
Find the length of DC.
Incorrect
Correct Answer:
4
Question 2
Calculate the size of the unmarked angle:
Incorrect
Correct Answer:
20
Question 3
Can a plane angle be found in a triangle?
Incorrect
Correct Answer:
No
Exercise 4
Assignment:
Given the parallel lines.
Find the angle α
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.
α+γ+δ=180°
α+60°+90°=180°
α+150°=180°
α=180°−150°
α=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 ∢UCH and angle ∢ACE are opposite angles.
ACE=ICH=2X
∢DAC and angle ∢ACE are collateral angles.
2x+DAC=180
DAC=180−2x
∢FGA and angle ∢DAB are opposite angles.
FGA=DAB=x−10
BAC=DAC−DAB=
180−2x−(x−10)=
190−3x
The sum of the angles in the triangle is 180
ACB+CAB+B=180
ACB=180−(190−3x)−(3x−30)=20
ACB=BAC
20=190−3x
x=56.67
Answer
56.67
Check your understanding
Question 1
Can a triangle have a right angle?
Incorrect
Correct Answer:
Yes
Question 2
Can a triangle have two right angles?
Incorrect
Correct Answer:
No
Question 3
DB is a side in triangle ABC
Incorrect
Correct Answer:
Not true
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
AE
Exercise #2
In an isosceles triangle, the angle between ? and ? is the "base angle".
Step-by-Step Solution
An isosceles triangle is one that has at least two sides of equal length. The angles opposite these two sides are known as the "base angles."
The side that is not equal to the other two is referred to as the "base" of the triangle. Thus, the "base angles" are the angles between each of the sides that are equal in length and the base.
Therefore, when we specify the angle in terms of its location or position, it is the angle between a "side" and the "base." This leads to the conclusion that the angle between the side and the base is the "base angle."
Therefore, the correct choice is Side, base.
Answer
Side, base.
Exercise #3
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
Exercise #4
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
No
Exercise #5
ABC is an isosceles triangle.
AD is the median.
What is the size of angle ∢ADC?
Video Solution
Step-by-Step Solution
In an isosceles triangle, the median to the base is also the height to the base.