Do parallel lines form a triangle?

Question

One definition of a triangle reads 'a polygon with 3 vertices and 3 edges' (https://en.wikipedia.org/wiki/Triangle). It occurred to me recently that an axiomatic and more information-rich definition might be more along the lines of 'the intersection of the interiors of two distinct angles', where the interior of an angle (taken from Greenburg's 'Euclidean and Non-Euclidean Geometry) is 'Given an angle $\angle CAB$ define a point D to be in the interior of $\angle CAB$ if D is on the same side of $\overline{AC}$ as B and if D is also on the same side of $\overline{AB} $ as C. (Thus, the interior of an angle is the intersection of two half-planes)'. I believe this is a neutral geometry definition. Intriguingly to me, not necessarily in neutral geometry, if one considers 'straight angles' as legitimate (side question, "do we?"), this would for example mean that the region enclosed by two parallels would technically be a triangle. So, I guess there are at least a couple questions here. In neutral geometry, Is a straight angle ($\pi$ radians if you like, or $\angle ABC$, where A*B*C e.g. B is between A and C), an angle just like any other angle? Is the region between parallel lines (not necessarily neutral geometry here) a triangle? Is the area defined and is it non-zero, zero, or infinite?