## Interior Angles of Polygons

Here’s another illustration:

## Triangles

The sum of the interior angles of a triangle equals 180°. Let’s try a triangle for a change: 180° is equal to 90° plus 60° plus 30°. It is effective in this triangle. Tilt a line by 10 degrees now: 180° is equal to 80° plus 70° plus 30°. It is still operational! One angle increased by 10°, while the other decreased by 10°.

## Quadrilaterals (Squares, etc)

(A quadrilateral is a shape with four straight sides.) Let’s attempt a square this time: 360° is equal to 90° plus 90° plus 90° plus 90°. The sum of a square is 360 degrees. Tilt a line by 10 degrees now: 80° plus 100° plus 90° plus 90° equals 360°. It still adds up to a whole 360 degrees. Quadrilaterals have interior angles that sum up to 360 degrees.

### Because there are 2 triangles in a square.

The internal angles of a triangle sum up to 180°. and the interior angles of a square add up to 360°. because a square may be formed by joining two triangles together!

## Pentagon

Because a pentagon has five sides and can be constructed from three triangles, the internal angles of a pentagon sum up to three 180° angles, or 540°. If the pentagon is regular (all angles are the same), then each angle is equal to 540°/ 5 = 108° (Example: verify that each triangle here adds up to 180°, and check that the pentagon’s inner angles sum up to 540°). All of the inside angles of a Pentagon come together to equal 540°.

## The General Rule

If we add a side (a triangle to a quadrilateral, a quadrilateral to a pentagon, and so on), we increase the total by another 180 degrees: As a result, the general rule is as follows: The sum of interior angles equals (n2) 180 degrees. For each angle (of a Regular Polygon), the formula is: (n 2/180 °/n Perhaps the following illustration will be of assistance:

### Example: What about a Regular Decagon (10 sides)?

In the sum of interior angles, (n 2) 180 degrees equals (102) 180 degrees equals 8 180 degrees equals 1440 degrees. And for a Regular Decagon, the following is true: Each inner angle is equal to 1440/10 =144°. Please keep in mind that inside angles are frequently referred to as “Internal Angles.”

## Why Does a Triangle Have 180 Degrees?

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## Types of triangles

To refresh your memory, there are three different sorts of fundamental triangles. They have the properties of being equilateral, isosceles, and scalene.

- An equilateral triangle is a triangle with three sides of equal length. It has two sides that are the same length as the other and one side that is a different length from the other two sides of the triangle
- A scalene triangle is a triangle with three sides that are all different lengths

The length of a triangle’s side has a direct impact on the angles of the triangle. What is the Pascal’s Triangle, and how does it work?

## Interior and exterior angles

One last piece of geometric language to cover before we go too far into our tale about triangles and the total number of degrees in their three angles is important to understand. In other words, the difference between an interior angle and an exterior angle is that the internal angle is smaller. The most straightforward approach to explain the distinction between these two concepts is to use an example. Because the triangle is the focus of today’s discussion, let’s speak about the inner and exterior angles of a triangle.

In other words, they’re the kinds of aspects that we’ve been discussing for quite some time.

As a result, when any of the internal angles of a triangle are added to their nearby outside angles, the result is always 180 0, which corresponds to a straight line.

## How many degrees is a triangle?

It is for this reason that the interior angles of a triangle always sum up to 180 0, which is the topic of today’s discussion. As it turns out, thinking about the interior and exterior angles of a triangle can help you find out the answer to this. To see what I’m talking about, either grab your imagination or a piece of paper because it’s time for a little mathematical arts and crafts project. First, sketch down an equal-sided right triangle with a single horizontal and one vertical leg, and with the hypotenuse extending diagonally down from the top left to the bottom right.

- Finally, build yet another duplicate of the original triangle and relocate it to the right so that it sits just next to the freshly formed rectangle that you just created.
- If this is the case, your photo should appear somewhat like this: What exactly is the aim of this photograph?
- Take a look at the two angles that make up the outer angle for that corner of the triangle (the ones labeled “B” and “C”) to see how they relate to one another.
- Furthermore, our little diagram demonstrates that the exterior angle in question is equal to the sum of the other two angles in the triangle.
- CONNECTED: Trigonometry and Its Applications We’ve chosen to use a right triangle in our illustration for the purpose of simplicity and clarity.
- To see this for yourself, try sketching a few pictures starting with different triangles of your choosing and seeing what happens.
- This is proven by our gorgeous and elegant little drawing, which we hope you like.

## Do all triangles equal 180 degrees? Can triangles have more?

Is it possible that there is a constraint to our drawing that is preventing us from seeing a more exotic possibility that we should be seeing? Here’s something to think about or do if you have the time. Obtain an uninflated balloon and place it on a level surface. Use a pencil to draw a triangle on the balloon that is as near to flawless as possible. If you have a protractor on hand, it would be beneficial to measure and add up the inner angles of the triangle to ensure that they are somewhat near to 180 degrees.

Was it ever resurrected?

How did this money end up in the wrong hands?

What does all of this have to do with the subject of whether the inner angles of a triangle always sum up to 180 degrees, as we appear to have discovered, and what does it mean?

The good news is that I know the solution to your question. Keep an eye out for next week’s post, in which we will begin our exploration of the odd and beautiful realm known as non-Euclidean geometry.

## Triangle Sum Theorem – Explanation & Examples

Different triangles have varying angles and side lengths, but one thing is constant: each triangle is formed of three interior angles and three sides, which can be the same length or of varying sizes and lengths, respectively. Consider the right triangle, which has one angle that is exactly 90 degrees and two sharp angles on either side. Isosceles triangles have two equal angles and two equal side lengths, which makes them a perfect triangle. Equilateral triangles having the same angles as well as the same side lengths as one another.

Despite the fact that the angles and side lengths of each of these triangles vary, they all follow the same principles and possess the same qualities.

- The Triangle Sum Theorem
- The interior angles of a triangle
- And how to utilize the Triangle Sum Theorem to obtain the interior angles of a triangle are all topics covered in this chapter.

## What is the Interior Angle of a Triangle?

The internal angles of a triangle are the angles that are created within a triangle, as defined by geometrical principles. The following are the characteristics of interior angles:

- It is known as the Triangle Angle Sum Theorem that the sum of internal angles equals 180 degrees. In a triangle, all of the internal angles are greater than 0° but less than 180°
- Throughout a triangle, the bisectors of all three internal angles meet at a point known as the in-center, which is located at the center of the triangle’s in-circle. When you add together all of the inner angles and outer angles, you get 180° (straight line).

## What is the Triangle Angle Sum Theorem?

Triangles have the property that their three internal angles total up to 180 degrees, which is a frequent attribute. This gets us to the Triangle Angle Sum Theorem, which is a very significant theorem in geometrical calculations. The Triangle Angle Sum Theorem states that the sum of the three internal angles of a triangle is always 180°. This may be expressed as:a + b + c = 180°

## How to Find the Interior Angles of a Triangle?

The Triangle Angle Sum Theorem may be used to find the third interior angle of a triangle when the first two interior angles of a triangle are known. To get the third unknown angle of a triangle, subtract the total of the first two known angles from 180 degrees and multiply the result by three. Now, let’s look at a couple of examples of problems: Exemple No. 1 Triangle ABC has the properties that A = 38° and B = 134°. Calculate the value of C. Solution As a result of the Triangle Angle Sum Theorem, we have: A + B + C = 180 degrees; 38° + 134° + Z = 180 degrees; 172° + C = 180 degrees.

- As a result, C = 8 degrees.
- Solution According to the Triangle Angle Sum Theorem (the sum of inner angles equals 180°), x + x + 18°= 180° via the Triangle Angle Sum Theorem Simplify by grouping together concepts that are similar.
- Calculate the difference between the two sides by 18° 2x + 18° – 18° = 180° – 18° 2x = 162°.
- Example 3Find the angles that are missing from the triangle shown below.
- Solution Let us suppose that 1 STangle = x°.
- In accordance with the Triangle Angle Sum Theorem, x° plus (x + 15) ° plus (x 15 + 66) ° equals 180°.
- ⇒ 3x + 81° = 180° 3x = 180° – 81° 3x = 99x =33° 3x = 99x =33° 3x = 99x =33° 3x = 99x =33° 3x = 99x =33° Now, in each of the three equations, substitute x = 33°.

three-dimensional RDangle = (x 15 + 66) degrees equals 33 degrees plus 15 degrees plus 66 degrees equals 81 degrees.

Exemple No.

Solution An additional angle (y °) is formed by adding the angles (2x + 10) °.

So the equations are as follows: y = 180° + (2x + 10) = 180° and y = 170°.

(2)Solve the two simultaneous equations by substitution: y = 170° – 2x = 115°; x + 170° – 2x = 115°; (ii) -x = 115° -170°x = 55° -x = 110° -170°x = 55° In contrast, y = 170° – 2x= 170° – 2(55) °y = 60°y = 170° – 110° As a result, the angles that are lacking are 60° and 55°.

6 To find out how many degrees an angle is, consider the triangle with three angles of 0°, (x + 20)°, and (2x + 40°).

x = 180° (sum of interior angles).

180° is equal to x + x + 2x + 20° + 40° 4x plus 60° equals 180° Subtract 60 from both sides of the equation.

4×4 = 120°/4x = 30° 4x = 120° Because of this, the triangle has three angles of 30 degrees, 50 degrees, and 100 degrees.

7 Fill in the blanks with the angles shown in the picture below.

BAD = x° DBC = x° DCB = 50° BAD = x° DBA = x° DBC = x° DBA = x° DBA = x° DBC = x° DBA = x° In this case, 50° plus 50° plus BDC = 180°BDC = 180° – 100°BDC = 80°50° plus 50° plus BDC = 180°BDC = 180° – 100°BDC However, z° plus 80° equals 180°.

(Angles on a straight line) As a result, z = 100°. In the triangle ADB, the following is written: 180° is equal to z° plus x plus x. 180° is equal to 100° plus 2x. 2x = 180 – 100 degrees the square root of two is 80 degrees while the square root of four is forty degrees

## Interior Angles Of A Triangle (video lessons, examples, step-by-step solutions)

Related Pages Exterior Angles Of A Triangle Types Of Triangles Angles In A Triangle The following diagram shows that the sum of angles in a triangle is 180°. Scroll down the page for more examples and solutions on how to find missing angles in a triangle.

### Interior Angles

Pages that are related Angles on the outside of a triangle Triangles Come in Many Shapes Angles in a triangle are defined as follows: The following picture illustrates that the sum of the angles of a triangle is 180° in the given situation. Continue reading for additional examples and solutions on how to discover the missing angles in a triangle by using the steps below.

### Properties of Interior Angles

- Whenever a triangle has three internal angles, the total of those angles is always 180°. Due to the fact that the inside angles sum up to 180°, every angle must be less than 180°.

### Find missing angles inside a triangle

For example, in the following triangle, determine the value of x. Solutions are as follows: x 24° + 32° = 180° (the total of the angles equals 180°). In this case, x + 56 degrees = 180 degrees; in this case, x = 180 degrees–56 degrees=124 degrees. In a triangle, how to locate the missing angle is explained. For example, in a triangle, find the angle that is missing. A = 15°, B =?, and C = 92° are the coordinates.

### Using Interior Angles Of Triangle To Set Up Equations

The sum of the internal angles of a triangle can be used to address difficulties in a variety of situations. Example:

- Calculate the unknown by writing an equation and solving for it
- Fill in the blanks with your response in each expression to find the measure of the angles
- Provide justifications for your responses

In order to find the unknown, build an equation using the sum of the interior angles of a triangle and solve for the unknown. You can practice many arithmetic concepts by using the freeMathway calculator and problem solution provided below. Make use of the examples provided, or put in your own issue and cross-reference your answer with the step-by-step instructions. If you have any suggestions, comments, or concerns regarding this site or page, please let us know. Thank you for taking the time to provide comments or inquiries on ourFeedbackpage.

## r/math – Stupid question: why do the angles of a triangle add up to 180 deg?

This is something that I have merely assumed to be true since third grade; but, why is it that the inner angles of a (euclidean) triangle add up to 180 degrees in the first place? I mean, it makes logical for the total of a rectangle to be 360 since, by definition, a rectangle has four 90-degree angles. However, in a triangle, all of the angles are flexible; thus, why is the sum of their angles constrained in this way? I tried to think about it logically, but the only thing that comes to me when I think of 180 degrees is a straight line.

- But it doesn’t make any sense; why would squeezing the line’s original angel save it from becoming distorted?
- The bigger the number of pinches you use, the greater the inner angle you obtain.
- Consequently, I present my request to you: could someone please assist me with this embarrassingly simple question?
- Please don’t be offended if I don’t respond to your message.
- As for the sheer number of them, I wanted to express my gratitude in a broad way because there is no way I am going to be able to go through each and every one of them.

There were some excellent comments that provided me with valuable information and assisted me in resolving this dilemma. Thank you very much.

## Triangle interior angles definition – Math Open Reference

The angles on the inside of the triangle are referred to as the internal angles. Consider the following: To reshape the triangle, drag the orange dots on each of the three vertices (A, B, and C). It is important to note that the internal angles always sum up to 180°. All of the inner angles of a triangle add up to a total of 180°. This means that only one of the angles can be 90 degrees or more. Because one angle in a right triangle is always 90 degrees, the other two angles in a right triangle must always sum up to 90 degrees.

For information on the characteristics of the interior angles of a polygon with any number of sides, see interior angles of a polygon with any number of sides.

## Properties

- It is always possible to add up the inner angles of a triangle to 180°
- Because the internal angles always sum up to 180°, every angle must be smaller than 180°
- Otherwise, the interior angles are useless. At a point, known as theincenter, the bisectors of each of the three internal angles come together
- This point also happens to be located at the center of the triangle’s incircle.

### Note

The inner angles of a triangle only add up to 180° when the triangle is planar, which means it is laying on a flat surface such as a table. If the triangle is not planar, for example, if it is resting on the curved surface of a sphere, the angles do not add up to 180° as they would otherwise.

## Other triangle topics

- Defining a triangle, the hypotenuse of the triangle, triangle inner angles, triangle external angles, triangle exterior angle theorem, Pythagorean Theorem, and Proof of the Pythagorean Theorem pythagorean triples
- A triangle’s circumcircle and incircle
- A triangle’s medians and altitudes
- The mid-section of a triangle
- The Triangle inequality
- The side / angle relationship

### Perimeter / Area

- A triangle’s perimeter
- A triangle’s surface area
- A triangle’s circumference Heron’s formula is as follows: The area of an equilateral triangle
- The area calculated using the “side angle side” technique
- The area calculated using the “side angle side” method
- A triangle’s surface area when the triangle’s perimeter is set

### Triangle types

- Obtuse triangle
- Acute triangle
- Right triangle
- Isosceles triangle
- Scalene triangle
- Equilateral triangle
- Equiangular triangle
- Isosceles trigonometry 3-4-5 triangle
- 30-60-90 triangle
- 45-45-90 triangle
- 3-4-5 triangle

### Triangle centers

- A triangle’s orthocenter is located at the intersection of the circumcenter and the incenter. a triangle’s incenter is located at the intersection of the circumcenter and the incenter.

### Congruence and Similarity

- Triangle type quiz, Ball Box problem, How Many Triangles, Satellite Orbits, and more related topics

(C) 2011 Copyright Math Open Reference. All rights reserved. All intellectual property rights are retained.

## Proof that a Triangle is 180 Degrees [Video]

It was one of the first things we all learned about triangles when we were children: the sum of the inner angles equals 180 degrees. You may have utilized this information to identify the missing angle in a triangle when you already knew the other two, and everything would have worked out fine for you. However, it is possible that a germ of skepticism or inquiry has slipped in. What is the source of the knowledge that the total of the angles is always 180 degrees? Is there any way for us to be certain that this is the case?

It is necessary to establish several fundamental truths about angles before we can mathematically show that the angles of a triangle will always add up to 180 degrees:

### Angles of a Triangle

The first item we need to go over is what a straight angle is and how it is defined. A straight angle is just a straight line, which is where the term “straight angle” comes from. To illustrate the three angles of a triangle, we’ve placed three points on the graph paper. Straight angle ABC has a 180-degree angle as its tangent. This will be useful in the future, so keep it handy. For the sake of seeing our next angles, let’s take two straight angles and have another line pass across them as follows: This is referred to as a transversal in our field.

- Observing the space between the parallel lines, we can observe that the sum of the two angles on either side of the transversal line equals 180 degrees.
- The fact that the straight line is a straight angle means that when it is divided in half, the two parts must sum up to the original measurement.
- It is possible to form a triangle by drawing one additional line that cuts over the parallel lines.
- The group of angles on the bottom left has remained unaltered, and a new group of angles has been produced by the new line crossing the bottom parallel line at the bottom.
- Now we can see that the three angles inside the triangle (B, EF, and EB) all sum up to 180 degrees as well.
- Angle C and Angle F are congruent since they both have the same value.
- Thus, the total of angles A, B, and C must equal the sum of angles B, E, and F.

The sum of angles A, B, and C must equal the sum of angles B, E, and F. Moreover, since the total of angles A, B, and C is known to be 180 degrees, it follows that the sum of angles B, E, and F must be 180 degrees as well. Presented below is a table that clearly outlines everything:

Statement | Reason |
---|---|

(m angle A+m angle B+m angle C=180 ^ ) | Definition of straight angle |

(m angle A+m angle E) | Alternate interior angles of transversal congruent if parallel lines |

(m angle C+m angle F) | Alternate interior angles of transversal congruent if parallel lines |

In order to understand what a straight angle is, we must first study what it is. A straight angle is nothing more than a straight line, which is where the term “straight angle” originates from. To depict the three angles of a triangle, we’ve placed three points on the graph paper to symbolize it. The straight angle ABC is equal to 180 degrees. When it comes to the future, this will be critical. To view our next angles, let’s take two straight angles and draw a line between them, as shown in the diagram: Transversals are what we refer to as such.

- The two angles on either side of the transversal line sum up to 180 degrees, which can be seen if we look between the parallel lines.
- When you cut an angle in half, the two halves must equal the original measure since a straight line is a straight angle.
- In order to form a triangle, we need to draw one more line that crosses the parallel lines.
- The group of angles on the bottom left has remained unaltered, and a new group of angles has been produced by the new line crossing the bottom parallel line on the bottom right.
- Now we can see that the three angles inside the triangle (B, EF, and EB) all sum up to 180 degrees, which is the answer.
- Due to the same reason, Angle C and Angle F are congruent.
- Thus, the total of angles A, B, and C must equal the sum of angles B, E, and F.
- Moreover, because the total of angles A, B, and C is known to be 180 degrees, it follows that the sum of angles B, E, and F must be 180 degrees as well.

## Review

Okay, let’s go over a couple of short review questions before we get started! 1. What is the length of a straight angle in meters? 2. If two angles are alternative interior angles of a transversal with parallel lines, then implies that the angles are likewise alternate exterior angles of a transversal with parallel lines. That concludes this review.

Thank you for reading! Thank you for taking the time to watch, and good luck with your studies!

## Frequently Asked Questions

In a straight angle, such as (A+B+C) in the red circle, the three angles combine to make a right angle of (180°). The transversals formed by the triangle’s side lengths establish angle pairs that are congruent with one another, as seen in the diagram. For example, the angles (A) and (A) are congruent because they are opposite interior angles of the same triangle. The interior angles of (B) and (B) are also congruent since they are both alternative interior angles. Due to the fact that (A=A) and (B=B), we may deduce that the internal angles (A+B+C) must likewise equal (180°).

#### Q

A straight angle, or simply a straight line, is defined as one that is (180°) from the horizontal. It is necessary for the internal angles of a triangle to add up to 180 degrees, which means that none of the angles may be exactly 180 degrees on their own.

#### Q

Whenever a triangle has three internal angles, the total of the three interior angles will always be (180°). A triangle cannot have an individual angle measure of (180°) since the other two angles would not exist if the angle measure of a triangle were ((180°+0°+0°)). The three angles of a triangle must be added together to equal (180°).

#### Q

When a triangle is constructed, the angle total will always equal (180°). In a quadrilateral, the total of the angles is equal to (360°), and a triangle may be formed by slicing a quadrilateral in half diagonally from corner to corner (see figure). Because a triangle is effectively half the size of a quadrilateral, the angle measurements of a triangle should be half the size of a quadrilateral as well. The half of (360°) is equal to (180°). Examples include the quadrilateral below, whose angles total (360°) since it is composed of four (90°) angles, and the triangle below, whose angles total (360°).

#### Q

An angle of (180°) will always result in a straight line when measured in degrees. This line is also known as a straight angle in some circles. It is possible to demonstrate the fact that a straight angle is (180°) by joining two right angles together. Two (90°) angles will combine to make a (180°) angle, which is a straight line.

## The Interior Angles of a Triangle Sum to Pi – Wolfram Demonstrations Project

Getting the live version up and running Copying from the download to the desktop

- Open in the Cloud
- Download to the Desktop
- Copy the Resource Object
- Open in the Cloud

Fullscreen It is necessary to have a Wolfram Notebook System. Interact with the freeWolfram Player or otherWolfram Language products on your PC, mobile device, or in the cloud. Any plane triangle’s internal angles can be combined to make a straight line if they are of the same size. An example of this is the blue triangle, which has one vertex fixed at the origin and two other vertices colored in yellow and green, as seen in the diagram below. Using the two gray auxiliary triangles, which are congruent with the blue triangle, a straight line through the origin may be drawn by connecting them.

Make use of the slider to make the relationships between the various angles more obvious. John Custy has contributed to this article (November 2011) Open work licensed under the Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License

## Snapshots

- Interior Angles of a Triangle
- Angles of a Triangle that total 180 degrees
- Total Angles of a Triangle

## Permanent Citation

Published on November 28th, 2011, John Custy’s “The Interior Angles of a Triangle Sum to Pi” Wolfram Demonstrations Project.