Many high school geometry textbooks define an angle as simply
the union of two rays with a common endpoint
The advantage of this definition is its simplicity. Among its disadvantages:
- It does not serve well for capturing the idea of a "direction": That is, there is no way to distinguish between a clockwise and a counterclockwise rotation.
- It more or less forces you to define "angle measure" in such a way that it is bounded between $0°$ and $180°$ (or between $0$ and $\pi$ radians).
However, the second objection can be remedied somewhat by introducing the notion of an angular region, defined as
a subset of the plane bounded by angle
One then has that every angle divides the plane into two angular regions. If the angle is not a straight angle, then those regions can be distinguished as the "interior" and "exterior" of the angle (by using betweenness as a criterion). You can then define the measure of an angular region according to the two cases: an interior angular region has the same measure as the angle that bounds it, and an exterior angular region has measure equal to $360°$ minus the measure of the angle that bounds it.
With this machinery in place, you can distinguish between a $270°$ and a $90°$ angular region. But you still can't capture the idea of direction. Nor can you discuss angles with negative measure, or angles with measure larger than $360°$. "Angular regions" are essentially static objects; if you want to capture the idea of "rotation" they will not work well for you.
(Edited to add: The approach taken above is more or less identical to the definition that followed in Geometry by Lang & Murrow, https://books.google.com/books?id=ntA5AlD3p4AC&printsec=frontcover#v=onepage&q&f=false. Thanks to user @whatever for providing a link to it in the comments.)
Having said all of that, though, I think it is worth rethinking the premise of this question. I do not know your students, but I strongly doubt that "rigor" is their main criterion for deciding how to trust you and how motivated to be. Remember: They do not know, yet, what rigor is, and they are not expecting it. They probably want a teacher who is clear and considerate, not someone who is rigorous and precise.
Note that what I have just written is not an argument against rigor. I think there are good reasons to be rigorous in a high school mathematics classroom -- but I am not sure you have named one.
You also might want to consider the proposition that "rigor" is not an absolute criterion but a relative one. Some definitions are more rigorous than others; typically the more rigorous, the less clear a definition is. The question you need to ask yourself is not "What is a rigorous definition for high school students?" but "How rigorous can I be without sacrificing clarity and confusing my audience?"
Supplement: As Ben Crowell points out, the answer above define "angle", but does not really address the question of angle measure. In most contemporary high school geometry curricula (at least in the United States) this is handled by virtue of something called a "Protractor Postulate", which asserts (in slightly more formal terminology than would be used in high school) the existence of a mapping that assigns to each angle $\angle ABC$ a real number, denoted $m\angle ABC$, with $0 \leq m\angle ABC \leq 180$ (if using degrees). Depending on the definition of "angle", the upper bound might be $360$; also, again depending on how "angle" is defined, the upper and lower bounds may or may not be strict.
The properties of the mapping $\angle ABC \mapsto m\angle ABC$ are further specified by additional postulates and/or definitions. One important property that we need is
Two angles are congruent if and only if they have the same measure
Whether the above is a postulate, a definition, or a consequence depends a lot on how the rest of the theory is structured -- in particular whether "congruent angle" is taken as an undefined relation, or defined in a measurement-free way, or whether the above is taken as the definition of congruent angles. For the sake of this discussion, let's call it the "Angle Congruence Postulate".
Another property we need is an "Angle Addition Postulate", which can be stated as saying that
Let three distinct rays have a common vertex. Then any two of the rays define two angular regions, with the third ray lying in exactly one of those regions and dividing it into two subregions. In this situation, the sum of the measures of the two subregions is equal to the measure of the full angular region.
Note that the Protractor Postulate, Angle Congruence Postulate and Angle Addition Postulate don't actually define the measure function, but they do characterize it, at least enough to make possible everything you need to do high school geometry.
A historical note: As far as I know the approach above was first introduced by Birkhoff & Beatley in Basic Geometry (1941), but was largely ignored for twenty years or so, until the School Mathematics Study Group (SMSG) adopted the "Ruler & Protractor Postulates" as part of the axiomatic framework in the New Math era Geometry textbooks, and it has remained the standard approach ever since. In the introduction to the first volume of the SMSG Geometry text, the editors wrote:
The basic scheme in the postulates is that of G. D. Blrkhoff. In this scheme, it is assumed that the real numbers are known, and they are used freely for measuring both distances and angles… It has been correctly pointed out that Euclid’s postulates are not logically sufficient for geometry, and that the treatments based on them do not meet modern standards of rigor. They were improved and sharpened by Hilbert. But the foundations of geometry, in the sense of Hilbert, are not a part of elementary mathematics, and do not belong in the tenth-grade curriculum. If we assume the real numbers, as in the Birkhoff treatment, then the handling of our postulates becomes a much easier task, and we need not face a cruel choice between mathematical accuracy and intelligibility. (Allen et. al., 1965, p. 10).
The last two sentences are perhaps most relevant for any one considering the question of how to bring mathematical rigor to the secondary classroom without sacrificing clarity.