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!

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 $A$ 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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Question 1

Is the straight line in the figure the height of the triangle?

Question 2

Is the straight line in the figure the height of the triangle?

Question 3

Can a plane angle be found in a triangle?

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

Is the straight line in the figure the height of the triangle?

Question 2

Is the straight line in the figure the height of the triangle?

Question 3

Is the straight line in the figure the height 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

Is the straight line in the figure the height of the triangle?

Question 2

Is the straight line in the figure the height of the triangle?

Question 3

According to figure BC=CB?

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

Exercise #1

Is the straight line in the figure the height of the triangle?

Video Solution

Step-by-Step Solution

The triangle's altitude is a line drawn from a vertex perpendicular to the opposite side. The vertical line in the diagram extends from the triangle's top vertex straight down to its base. By definition of altitude, this line is the height if it forms a right angle with the base.

To solve this problem, we'll verify that the line in question satisfies the altitude condition:

Step 1: Identify the triangle's vertices and base. From the diagram, the base appears horizontal, and the vertex lies directly above it.
Step 2: Check the nature of the line. The line is vertical when the base is horizontal, indicating perpendicularity.
Conclusion: The vertical line forms right angles with the base, thus acting as the altitude or height.

Therefore, the straight line depicted is indeed the height of the triangle. The answer is Yes.

Answer

Yes

Exercise #2

Can a triangle have a right angle?

Video Solution

Step-by-Step Solution

To determine if a triangle can have a right angle, consider the following explanation:

Definition of a Right Angle: An angle is classified as a right angle if it measures exactly $90^\circ$ .
Definition of a Right Triangle: A right triangle is a type of triangle that contains exactly one right angle.
According to the definition, a right triangle specifically includes a right angle. This is a well-established classification of triangles in geometry.

Thus, a triangle can indeed have a right angle and is referred to as a right triangle.

Therefore, the solution to the problem is Yes.

Answer

Yes

Exercise #3

Is the straight line in the figure the height of the triangle?

Video Solution

Step-by-Step Solution

To determine if the straight line is the height of the triangle, we'll analyze its role within the triangle:

Step 1: Observe the triangle and the given line. The triangle seems to be made of three sides and a vertical line within it.
Step 2: Recall that the height of a triangle, in geometry, is defined as a perpendicular dropped from a vertex to the opposite side.
Step 3: Examine the positioning of the line: The vertical line starts at one vertex of the triangle and intersects the base, appearing to be perpendicular.
Step 4: Verify perpendicularity: Given that the line is shown as a clear vertical (and a small perpendicular indicator suggests perpendicularity), we accept this line as the height.
Step 5: Conclude with verification that the line is effectively meeting the definition of height for the triangle in the diagram.

Therefore, the vertical line in the figure is indeed the height of the triangle.

Yes

Answer

Yes

Exercise #4

Is the straight line in the figure the height of the triangle?

Video Solution

Step-by-Step Solution

In the given problem, we have a triangle depicted with a specific line drawn inside it. The question asks if this line represents the height of the triangle. To resolve this question, we need to discern whether the line is perpendicular to one of the sides of the triangle when extended, as only a line that is perpendicular from a vertex to its opposite side can be considered the height.

The line in question is shown intersecting one of the sides within the triangle but does not form a perpendicular angle with any side shown or the ground (as is required for it to be the height of the triangle). A proper height would typically intersect perpendicularly either at or along the extended line of the opposite side from a vertex.

Therefore, based on the visual clues provided and the typical geometric definition of a height (or altitude) in a triangle, this specific line does not fit the criteria for being a height.

Thus, we conclude that the line depicted is not the height of the triangle. The correct answer is No.

Answer

Exercise #5

Is the straight line in the figure the height of the triangle?

Video Solution

Step-by-Step Solution

The height of a triangle is defined as the perpendicular distance from a vertex to the line containing the opposite side (base). In this problem, we observe a vertical line segment drawn from a point on the base (horizontal line at the bottom of the triangle) to some level above the base. To determine if this line is a height, it must be perpendicular to the base and also reach to the opposite vertex of the triangle.

In the provided figure, the vertical line extends vertically from the base but does not connect to the opposite vertex of the triangle (at the top). Instead, it terminates at some intermediate point above the base. Since the line does not satisfy the full condition of being perpendicular and reaching an opposite vertex, it cannot be considered the height of this triangle.

Therefore, the given straight line is not the height of the triangle.

The correct and final answer is: No.

Answer

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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