Are angles of a triangle always positive?

Question

If a question says "$A,B,$ and $C$ are the angles of a triangle".

Can I assume $A,B,$ and $C$ to be positive and its properties such as arithmetic mean greater equal to geometric mean?

Also angle sum property of triangle seems to look flawed. I think it should be about absolute value of angle a triangle add up to π rather than sum of angles of triangle add up to π. Or maybe I am missing something. Thanks a lot.

Answer 1

There is a difference between the terms angle and oriented angle.

Planar geometry generally don't use oriented angles, only "common" angles between $0$ and $2\pi.$

So you may assume angles of triangle as positive values in the interval $(0, \pi).$

Answer 2

For angles, we have 4$\pi = 0$, so for example $3\pi = -\pi$. Thus, there is no natural way to talk about positive and negative angles. You can define it, but then you define a notation to stand for a positive and another notation for a negative angle, as this makes computation easier in some cases.
For triangles, we usually define that each angle should be between $0$ and $2\pi$, and thus they are always positive. There might be other definitions, e.g. when looking at angles in a triangle inside the unit circle, but again, these are just definitions useful in this particular case, it is never an overall property (of triangles).

Answer 3

In Euclidean geometry, angles are non-negative. This follows from whole-part postulate (the whole is greater than a part).

Answer 4

Any triangle splits the plane in two region: the interior and exterior of the triangle. Generally an angle of the triangle is assumed to be the internal angle between two sides. If not otherwise stated it assumed to be not oriented and thus positive.

The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other.

https://en.wikipedia.org/wiki/Angle#Measuring_angles