Has triangle an angle? I read axiomatic geometry and found the following definitions:

Points $A$ and $B$ and all those points that lie between those points is a line segment.

If $AB$ and $AC$ are two rays that does not belong to the same line, then the union of those rays forms an angle.

If three points are not collinear, they form a triangle. Line segments $AB$, $BC$ and $CA$ are the sides of a triangle.

Now, has a triangle an angle? I mean if triangle has line segments but not rays then I guess triangle has no angles unless we extend the line segments to the rays. Am I correct?
Yes, the angles in a triangle are defined from extending the segments into rays.
A triangle has... three angles! That's one definition of the triangle. 

However, there are degenerate triangles, when the three points are colinear, but all different one from the others. Then one angle is $\pi$ while the other tow are $0$.