# Triangle Sum Theorem Worksheet, Examples, and Definition

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### Key Points about Triangle Sum Theorem

• The Triangle Sum Theorem states that the sum of the three interior angles of a triangle is always equal to 180 degrees.
• The formula for finding the sum of the interior angles of a triangle is derived from the Triangle Sum Theorem.
• The Triangle Sum Theorem is a fundamental concept in geometry that is used to solve many problems involving triangles.

## Triangle Sum Theorem Definition

Triangle Angle Sum Theorem is mathematical proof about the interior angles of a triangle. The Triangle Angle Sum states that all the angles inside of a triangle must add up to 180 degrees. This means that you can use the Triangle Angle Sum to find a missing interior angle of a triangle by adding the two angles that you know together and then subtracting the sum from 180 degrees. Continue below to try some Triangle Sum Theorem Practice Problems or click the download links on the right to download a free Triangle Angle Sum Theorem Worksheet.

The Triangle Sum Theorem is a fundamental concept in geometry that states that the sum of the three interior angles of a triangle is always equal to 180 degrees. This theorem is essential to understanding the properties of triangles and is used in many mathematical proofs and applications. The theorem can be applied to any type of triangle, whether it is equilateral, isosceles, or scalene.

Using The Triangle Sum Theorem, mathematicians can determine the measure of an unknown angle in a triangle when the measures of the other two angles are known. This is done by subtracting the sum of the two known angles from 180 degrees. The formula for finding the sum of the interior angles of a triangle is also derived from the Triangle Sum Theorem.

The Triangle Sum Theorem is a powerful tool in geometry that is used to solve many problems involving triangles. By understanding this theorem, one can gain a deeper understanding of the properties of triangles and their angles. In the following sections, we will explore the formula for the sum of the interior angles of a triangle, provide examples of how to use the Triangle Sum Theorem, and answer some frequently asked questions about triangles and their angles.

Common Core Standard: 8.G.5
Related Topics: Pythagorean Theorem, Parallel Lines Cut by a Transversal, Exterior Angle of a Triangle, Volume of a Cylinder, Volume of a Cone, Volume of a Sphere

## Sum of Interior Angles of a Triangle Formula

The sum of interior angles of a triangle is a fundamental concept in geometry. The Triangle Sum Theorem states that the sum of the three interior angles of any triangle is always 180 degrees. This theorem is also known as the Angle Sum Property of a Triangle.

The Triangle Sum Theorem is a crucial property that helps in solving various problems related to triangles. It is used in many geometric proofs and is an essential tool in the study of geometry. The theorem is also helpful in determining the measure of an unknown angle in a triangle.

The formula for the sum of interior angles of a triangle can be derived from the Triangle Sum Theorem. The formula is given as:

``````Sum of Interior Angles of a Triangle = 180 degrees
``````

This formula can be used to find the measure of an unknown angle in a triangle when the measures of the other two angles are known. For example, if two angles of a triangle measure 60 degrees and 80 degrees, then the measure of the third angle can be found by subtracting the sum of the other two angles from 180 degrees.

In addition to the Triangle Sum Theorem, there are other formulas that can be used to find the measure of an interior angle of a triangle. One such formula is the Exterior Angle Theorem, which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

In conclusion, the sum of interior angles of a triangle is an important concept in geometry. The Triangle Sum Theorem is a fundamental property that is used in many geometric proofs and helps in solving various problems related to triangles. The formula for the sum of interior angles of a triangle is straightforward and can be used to find the measure of an unknown angle in a triangle. ## 3 Simple Triangle Sum Theorem Examples

The Triangle Sum Theorem is a fundamental concept in geometry that states that the sum of the three interior angles in a triangle adds up to 180°.

1. Add the two angles that you know together.
2. Take the sum of those two angles and subtract it from 180.
3. The resulting amount is the missing angle.

Here are some examples that illustrate this theorem:

### Example 1

Suppose a triangle has two angles with measures of 35° and 85°. What is the measure of the missing angle?

To solve this problem, use the Triangle Sum Theorem:

``````35° + 85° + x = 180°
``````

Simplify the equation by combining like terms:

``````120° + x = 180°
``````

Subtract 120° from both sides:

``````x = 60°
``````

Therefore, the missing angle has a measure of 60°.

### Example 2

Suppose a triangle has two angles with measures of 2x and 3x. What is the measure of the third angle?

To solve this problem, use the Triangle Sum Theorem:

``````2x + 3x + y = 180°
``````

Simplify the equation by combining like terms:

``````5x + y = 180°
``````

Suppose the value of x is 20. Then, substitute 20 for x in the equation:

``````5(20) + y = 180°
``````

Simplify the equation:

``````100 + y = 180°
``````

Subtract 100 from both sides:

``````y = 80°
``````

Therefore, the third angle has a measure of 80°.

### Example 3

Suppose a triangle has two angles with measures of 2x and 3x. What is the value of x?

To solve this problem, use the Triangle Sum Theorem:

``````2x + 3x + y = 180°
``````

Simplify the equation by combining like terms:

``````5x + y = 180°
``````

Suppose the value of y is 60. Then, substitute 60 for y in the equation:

``````5x + 60 = 180°
``````

Simplify the equation:

``````5x = 120°
``````

Divide both sides by 5:

``````x = 24°
``````

Therefore, the value of x is 24°.

## Angles in a Triangle Questions and Answers: 5 Quick Problems

/5

Triangle Sum Theorem Quiz

Click Start to begin the practice quiz!

1 / 5

Find x for the missing angle measure of the following triangle:

Angle 1: 23 degrees

Angle 2: 132 degrees

Angle 3: x degrees

2 / 5

Find x for the missing angle measure of the following triangle:

Angle 1: 67 degrees

Angle 2: x degrees

Angle 3: 98 degrees

3 / 5

Find x for the missing angle measure of the following triangle:

Angle 1: x degrees

Angle 2: 57 degrees

Angle 3: 117 degrees

4 / 5

Find x for the missing angle measure of the following triangle:

Angle 1: 45 degrees

Angle 2: 60 degrees

Angle 3: x degrees

5 / 5

Find x for the missing angle measure of the following triangle:

Angle 1: 85 degrees

Angle 2: 20 degrees

Angle 3: x degrees

0%

## Sum of Angles in a Triangle

The Triangle Sum Theorem, also known as the Angle Sum Property of a Triangle, states that the sum of the interior angles of any triangle is always equal to 180 degrees. This theorem is a fundamental concept in geometry and is used in various applications, including calculating unknown angles in a triangle.

To better understand the Triangle Sum Theorem, let’s consider a triangle with three vertices labeled A, B, and C. The interior angles of the triangle are labeled as ∠A, ∠B, and ∠C, respectively. According to the theorem, the sum of these angles is always 180 degrees, or ∠A + ∠B + ∠C = 180°.

The Triangle Sum Theorem can be proven in various ways, including using parallel lines and transversals, congruent triangles, and the Law of Cosines. One of the most common proofs involves drawing a line parallel to one side of the triangle and using the resulting parallel lines and transversals to find the sum of the interior angles.

The Angle Sum Property of a Triangle has many applications in geometry. For example, it can be used to find the measure of an unknown angle in a triangle when the measures of the other two angles are known. It can also be used to prove the congruence of triangles and to find the area of a triangle using the formula A = 1/2 bh, where b is the base of the triangle and h is its height.

In summary, the Triangle Sum Theorem is a fundamental concept in geometry that states that the sum of the interior angles of any triangle is always equal to 180 degrees. This theorem has many applications in geometry and is used to find unknown angles, prove the congruence of triangles, and find the area of a triangle.

## How many Angles in a Triangle?

A triangle is a three-sided polygon. It is a fundamental shape in geometry, and its properties have been studied for centuries. One of the most important properties of a triangle is the number of angles it has.

As per the Triangle Sum Theorem, the sum of the three interior angles in a triangle is always equal to 180 degrees. This means that no matter what type of triangle it is – equilateral, isosceles, or scalene – the sum of its angles will always be the same.

To understand this theorem better, consider an equilateral triangle. In an equilateral triangle, all three sides are of equal length, and all three angles are of equal measure. Since the sum of the angles in a triangle is 180 degrees, each angle in an equilateral triangle must be 60 degrees.

Similarly, in an isosceles triangle, two sides are of equal length, and two angles are of equal measure. The third angle is different. However, the sum of all three angles is still 180 degrees.

In a scalene triangle, all three sides are of different lengths, and all three angles are of different measures. Nevertheless, the sum of all three angles is still 180 degrees.

Knowing the number of angles in a triangle is essential for solving various geometry problems. The Triangle Sum Theorem is a fundamental concept that helps in understanding and solving such problems.

## What do Angles in a Triangle add up to?

The Triangle Sum Theorem, also known as the Angle Sum Theorem, states that the sum of the interior angles of a triangle is always equal to 180 degrees. This theorem is a fundamental concept in geometry and is used extensively in solving problems related to triangles.

The theorem applies to all types of triangles, whether they are equilateral, isosceles, or scalene. In an equilateral triangle, where all three sides are equal, each angle measures 60 degrees. In an isosceles triangle, where two sides are equal, the two angles opposite the equal sides are also equal. In a scalene triangle, where all three sides are different lengths, all three angles are different.

The Triangle Sum Theorem can be proven using various methods, including the method of drawing an auxiliary line. This method involves drawing a straight line that intersects one of the angles of the triangle and is parallel to the opposite side. By doing so, the triangle is divided into two smaller triangles, and the sum of the angles in each smaller triangle can be calculated. Since the sum of the angles in the larger triangle is equal to the sum of the angles in the two smaller triangles, the theorem is proven.

Knowing the Triangle Sum Theorem is essential in solving problems related to triangles. For example, if two angles of a triangle are known, the third angle can be calculated by subtracting the sum of the two known angles from 180 degrees. Similarly, if the lengths of two sides of a triangle and the angle between them are known, the length of the third side can be calculated using the Law of Cosines.

In conclusion, the Triangle Sum Theorem is a fundamental concept in geometry that states that the sum of the interior angles of a triangle is always equal to 180 degrees. It applies to all types of triangles and is used extensively in solving problems related to them.

## FAQ about Triangle Sum Theorem

### What is the sum theorem formula?

The sum theorem formula states that the sum of all three angles in a triangle is equal to 180 degrees. This formula is also known as the Triangle Sum Theorem.

### How do you solve the sum theorem of a triangle?

To solve the sum theorem of a triangle, you need to add up the measures of all three angles in the triangle. If the sum of the angles is equal to 180 degrees, then the theorem is satisfied.

### What Is the Triangle Sum Theorem in Geometry?

The Triangle Sum Theorem is a fundamental concept in geometry that states that the sum of all three interior angles of a triangle is always equal to 180 degrees. This theorem is important because it applies to all types of triangles.

### How to Prove the Triangle Sum Theorem?

The Triangle Sum Theorem can be proved using several methods, including algebraic proofs, geometric proofs, and trigonometric proofs. One of the most common methods is to draw a line parallel to one side of the triangle and use the properties of parallel lines and angles to prove that the sum of the angles is equal to 180 degrees.

### What is the exterior angle sum theorem?

The exterior angle sum theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. In other words, the exterior angle of a triangle is equal to the sum of the opposite interior angles.

### What is the triangle side sum theorem?

The triangle side sum theorem states that the sum of any two sides of a triangle is always greater than the third side. This theorem is important because it helps to determine whether a given set of side lengths can form a valid triangle.

### Do all triangles add up to 180?

Yes, all triangles add up to 180 degrees. This is because of the Triangle Sum Theorem, which states that the sum of all three interior angles of a triangle is always equal to 180 degrees.

### What is the sum of all 3 sides of a triangle?

The sum of all three sides of a triangle is equal to the perimeter of the triangle. The perimeter is the total length of all sides of the triangle.

## Triangle Sum Theorem Worksheet: Video Explanation

Watch our free video on how to solve Triangle Angle Sum Theorem. This video shows how to solve problems that are on our free Triangle Angle Sum worksheet that you can get by submitting your email above.

Watch the free Sum of Angles in a Triangle video on YouTube here: Triangle Sum Theorem

Video Transcript:
This video is about the triangle sum theorem. In order to show you some examples about the triangle sum theorem we’re going to use our worksheet that you can get on our website mathcation.com.

The first thing we need to know about the triangle sum theorem is the rule that all the angles in a triangle have to add up to 180 degrees. Angle 1 plus angle 2 plus angle 3 will always add up to 180 degrees. In the case of our first problem we are given 34 degrees 97 degrees and then we’re missing this angle here. We have to find this missing angle here.

The easiest way to do this is to add up these two. 34 which we’re going to say is angle 1 plus angle 2 which is 97 plus measure of angle 3, which we don’t know, is equal to 180 degrees. Now we’re going to add these two together 34 plus 97 is 131 and then we don’t know the measure of angle 3 and it’s 180 degrees. Then the last step is to subtract 180 minus 131 to get our answer of 49 degrees  the measure of angle 3 will be equal to 49 degrees.

Moving onto number 2 we’re given a new triangle and we’re given 24 degrees here and then we’re given a right angle here. All right angles are equal to 90 degrees. A shortcut to do this instead of doing the way we did in problem 1 would be to add these two together and then subtract it from 180 degrees.

The quickest way to do this would be 90 plus 24 which is equal to 114 degrees and then you just do 180 minus 114 because they have to add up to 180. 180 minus 114 if 66. This angle here has to be 66 degrees because all three angles must add up to 180. 66 plus 24 plus 90 equals 180. We know that’s our answer.

You can find other triangle sum theorem worksheet answers when you download the free triangle angle sum theorem worksheet. The free download includes a practice worksheet as well as the triangle sum theorem answer key.

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