Why does the sum of a triangle's angles always equal to 180 degrees?

Answer 1

In Euclidean Geometry:

May be the question is, if triangles go all the way around, why isn't it 360 degrees (full turn around) if your pencil comes back to the initial point when you draw one?

Well, first in a triangle it is impossible to have more than one angle at 90 degrees or more, thus at least two angles must be at a maximum of 60 degrees. It is these acute angles what let it be 180 degrees. And since the angles are linearly proportional. As one widens another one becomes more acute at the exact same rate if you hold the third one, and if you don't hold the third one the sum of the change of the other two will equal the change of the first one. So to keep this proportionality the sum has to always be 180.

Now the question becomes, have we found a way to turn half less (degrees) around if we turn in triangle shape rather than a circle or a square (360 degrees) when traveling.

In Reimannian Geometry (aka Elliptic Geometry):

Triangles do not necessarily add up to 180 degrees. For example, longitudinal lines on the Earth originate at the North Pole. They intersect the Equator at 90 degree angles. The lines proceed to the North Pole and intersect there at whatever angle measure (x). 90+90+x>180, assuming x>0, so in this form of Geometry, the triangles will always measure more than 180 degrees.