Math Myth: The sum of all angles in a triangle is 180°

In summary, the sum of all angles in a triangle being 180° is a fundamental property of triangles in Euclidean geometry, which allows for the formation of geometric proofs and serves as a basis for various mathematical concepts. This property always holds true in Euclidean geometry, including all types of triangles. However, in non-Euclidean geometries, there may be exceptions depending on the curvature of the space. The sum of all angles in a triangle is related to other geometric concepts such as the Pythagorean Theorem, the Law of Sines, and the Law of Cosines, and plays a crucial role in the study of polygons, circles, and trigonometry.
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Greg Bernhardt
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From @fresh_42's Insight
https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/

Please discuss!

We all live on a globe, a giant ball. The angles of a triangle on this ball add up to a number greater than ##180°##.

Kugeldreieck.png


And the amount by which the sum extends ##180°## isn't even constant. It depends on the size of the triangle. The flat triangle with angle-sum ##180°## is the exception, not the norm. The real world is crooked.
 
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  • #2
This is something I never knew in school! Has anyone else heard of this before?
 

What is the "Math Myth" about the sum of all angles in a triangle?

The "Math Myth" about the sum of all angles in a triangle is the belief that the sum is always exactly 180°. This is often taught in schools and is a common misconception.

Is it true that the sum of all angles in a triangle is always 180°?

No, this is not always true. The sum of all angles in a triangle can vary depending on the type of triangle and the geometry it is placed in.

What are some examples of triangles where the sum of all angles is not 180°?

Some examples include spherical triangles, where the sum can be greater than 180°, and hyperbolic triangles, where the sum can be less than 180°.

Why is it important to understand the concept of the sum of all angles in a triangle?

Understanding the concept of the sum of all angles in a triangle is important in geometry and other mathematical fields. It helps to determine the measurements and properties of triangles and can be applied to other shapes as well.

How can we prove that the sum of all angles in a triangle is not always 180°?

There are several mathematical proofs that can demonstrate that the sum of all angles in a triangle is not always 180°. One example is using the Pythagorean Theorem to show that in a right triangle, the sum of the two acute angles is always less than 90°, thus making the sum of all angles less than 180°.

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