# Can the sum of angles of a planar triangle be greater than 180?

• hermy
In summary, the sum of angles of a planar triangle is always exactly 180 degrees. However, on a sphere, the sum of angles may be slightly greater or lesser than 180 degrees, depending on the area of the triangle.
hermy
Can the sum of angles of a planar triangle be greater than 180??

hii

i read somewhere that the sum of the angles of a triangle is always a little bit greater than or lesser than 180 degrees? how true is that?

thank you

Welcome to PF!

Hi hermy! Welcome to PF!
hermy said:
i read somewhere that the sum of the angles of a triangle is always a little bit greater than or lesser than 180 degrees? how true is that?

It's true on the Earth's surface …

on a sphere, the sum of the angles minus 180º is proportional to the area of the triangle …

but it's not true on a plane: the sum is always exactly 180º.

## 1. Can the sum of angles of a planar triangle ever be greater than 180 degrees?

No, the sum of angles of a planar triangle will always be exactly 180 degrees. This is a fundamental property of Euclidean geometry and is known as the Triangle Sum Theorem.

## 2. Why can't the sum of angles of a planar triangle be greater than 180 degrees?

This is because the three angles of a triangle are always complementary, meaning they add up to exactly 180 degrees. This can be proven through various geometric proofs and is a fundamental principle in mathematics.

## 3. Are there any exceptions to the Triangle Sum Theorem?

No, this theorem holds true for all planar triangles, regardless of their size or shape. It is a universal property in geometry and has been proven mathematically.

## 4. Can the sum of angles of a non-planar triangle be greater than 180 degrees?

Yes, in non-Euclidean geometries such as spherical or hyperbolic geometry, the Triangle Sum Theorem does not hold true. In these geometries, the sum of angles of a triangle can be greater than or less than 180 degrees.

## 5. What are some real-world applications of the Triangle Sum Theorem?

The Triangle Sum Theorem is used in various fields such as architecture, engineering, and navigation. It helps in calculating the angles and measurements of triangular structures and is also used in GPS systems to determine the location and direction of travel.

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