Has triangle an angle?

Question

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?

Answer 1

Modern axiomatisation of Euclid's geometry is due to David Hilbert in his book Grundlagen der Geometrie (The Foundations of Geometry, 1899).

A modern treatment is in Robin Hartshorne, Geometry Euclid and Beyond (2000).

According to Hilbert [see page 4 of the English translation] :

Definition. We will call the system of two points $A$ and $B$, lying upon a straight line, a segment and denote it by $AB$ or $BA$.

Hilbert states [page 6] :

Definition. A system of segments $AB, BC, CD, \ldots, KL$ is called a broken line joining $A$ with $L$ and is designated, briefly, as the broken line $ABCDE \ldots KL$. The points lying within the segments $AB, BC, CD, \ldots, KL$, as also the points $A, B, C, D, \ldots, K, L$, are called the points of the broken line. In particular, if the point $A$ coincides with $L$, the broken line is called a polygon and is designated as the polygon $ABCD \ldots K$.

The segments $AB, BC, CD, \ldots, KA$ are called the sides of the polygon and the points $A, B, C, D, \ldots, K$ the vertices. Polygons having $3, 4, 5, \ldots, n$ vertices are called, respectively, triangles, quadrangles, pentagons, ..., $n$-gons.

Thus, triangles are identified by three non-collinear point $A, B, C$.

See also [page 6] :

All of the points of [a line] $a$ which lie upon the same side of [a point] $O$, when taken together, are called the half-ray emanating from $O$. Hence, each point of a straight line divides it into two half-rays.

Thus, considering the segment $AB$ on line $a$, we have that $A$ divides $a$ into two half-rays, one of which contains point $B$, and so all the segment $AB$.

Finally, we have [page 8] :

Definitions. Let be any arbitrary plane and $h, k$ any two distinct half-rays lying in and emanating from the point $O$ so as to form a part of two different straight lines. We call the system formed by these two half-rays $h, k$ an angle [...].

Thus, the half-rays containing segments $AB$ and $AC$ form an angle; the same for the couples $BA$ with $BC$ and $CB$ with $CA$.

Answer 2

Yes, the angles in a triangle are defined from extending the segments into rays.

Answer 3

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 $0$.