Rigorously defining the concept of an angle for high school students

Arriving at a rigorous definition of the concept of angle for high school students is not as easy as expected.

Google search provided me with many definition that are too technical or too vague IMO.

Here is my current version:

Measurement of the counterclockwise rotation of one ray from another ray with the same endpoint (vertex). Usually measured in degrees from $0$ to $360$ or radians from $0$ to $2\pi$.

I ask for your feedback. Does my definition fall (break) down in certain (relevant) settings? Does it obfuscate? Is there a better way to provide intution for the concepts.

One problem I see is that depending on the ray you decide has rotated you get different angles. For example how do I know if the angle is $90°$ or $270°$.

• Why do you want a rigorous definition of an angle? Is there something you want to achieve that an intuitive geometrical approach does not give? The quality of a definition depends on the purpose. – Joonas Ilmavirta Jan 3 '16 at 14:33
• i.stack.imgur.com/t1tNf.png – user5402 Jan 3 '16 at 16:25
• @mweiss books.google.com/… – user5402 Jan 3 '16 at 18:25
• @mweiss You're welcome. This approach is taken also in this book (in french). – user5402 Jan 3 '16 at 18:50
• What you give is not a definition of an angle. It's a definition of the measure of an angle which is a function that maps geometric objects (angles) to the interval $[0,2 \pi)$, or maps these objects (angles) to equivalence classes represented by elements of the interval $[0,2 \pi).$ You also need to define "rotation" first, which is problematic if your goal is simplicity. In my opinion, the vector definition is the best, and it is not too difficult for most 9th grade geometry students who discuss vectors in their geometry classes anyhow. – Andrew Jan 4 '16 at 21:18

Many high school geometry textbooks define an angle as simply

the union of two rays with a common endpoint

• 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.

• Agree mostly with all points. Will stick with rotation in there to prepare for the idea of negative angles. Union seems vague to me (not a native English speaker). – snoram Jan 3 '16 at 19:50
• "Union" is a precise notion from set theory, denoted $A \cup B$. Are you familiar with that symbol? – mweiss Jan 3 '16 at 19:57
• Yes, of course (from set theory): all elements that are in either A or B or both. But what are the elements of a ray? – snoram Jan 3 '16 at 20:15
• A ray is a set of points, as is a line, a segment, etc.. How else are you defining "ray"? – mweiss Jan 4 '16 at 0:26
• This doesn't really address the question of how to rigorously define a measure of angle. You've defined an angle, but you haven't defined its measure. – Ben Crowell Jan 4 '16 at 15:21

My answer will probably go deeper than a high-school student will be exposed to, but I think it is still relevant to a high-school teacher. The most conclusive part is far below in bold, if the rest is too long. Bottom line is, angles are tricky and easy to underestimate.

First, it may be preferable to distinguish the geometric object from the measurement. I will therefore use angle for the geometric object (see below) and angle measure for the measurement (in radians or degrees or whatever).

Next, it is important to realize that there are many different kind of angles, defined by e.g. pairs of incident (i.e. sharing their starting point) half-lines, pairs of vectors, pairs of intersecting lines, and the ordered version of each of those. You may want to go with angular sectors (aka component of the complement of a pair of incident half-lines), as mentioned in a comment.

Then, one can ask when to consider two angles equal (e.g., when defining an isosceles triangle). Usually, one defines an angle measure and defines equality of angles by equality of measures, but this makes very difficult to understand the difference between the different situations above: in fact the measures do not live in the set of real numbers but in a quotient set: as everyone knows, angle measures of $2\pi$ and $0$ are the same really (except in cases they are not, see below). But when it comes to ordered pairs of lines, in fact $\pi/3$ and $-2\pi/3$ are also the same (more on that later): the various situations lead to various quotient sets where the measurement has its meaning.

So, the universal (in the sense: for all the cases above, other approaches to the whole topic are possible) definition of equality of angles is when one can be mapped to the other by a displacement (aka direct isometry of the plane). For example for angles as unordered pairs of vectors, two are equal when there is a displacement mapping each of the vectors of the first pair to a vector of the second pair (disregarding which is mapped to which). I fact, I used the word "equality" because from a deeper, more formal point of view we define angles as of the geometric objects under the action of the group of displacements of the plane.

Then, one tries to capture this equivalence relation between angles by an angle measure. For an ordered pair of vectors, the usual definition is to consider the "angle" of the rotation mapping the first one to the second one. This seems a recursive definition, but is not when one has enough background: a rotation can be defined as a displacement with a fixed point, one proves that it is a affine map whose linear part has a matrix of the form $$\begin{pmatrix}\cos \theta & -\sin\theta \\ \sin \theta & \cos\theta\end{pmatrix}$$ for some number $\theta$ (in any direct orthonormal basis), and then calls $\theta$ the angle of the rotation. Of course, for this one needs to define $\cos$ and $\sin$, usually via analytic tools, and that's a moral problem to have everything patch up together.

When one considers angles defined by an ordered pair of intersecting lines, then there are two possible ways to get an ordered pair of vectors: when one takes one vector in the direction of each line, then changing exactly one into its opposite is just as valid a choice, and adds (or substracts) the angle measure by $\pi$. So one ends up working in $\mathbb{R}/\pi\mathbb{Z}$. Unordered pairs are even messier.

The easiest case (in which one have no oriented angles) is to consider angular sectors and use the Greek definition of angle measure: take a circle centered at the vertex of the sector, and divide the length of its intersection with the sector by its radius. The number obtained does not depend on the chosen circle, and is the most geometric possible definition of an angle measure.

This is basically the point of view proposed by mweiss, but with an actual definition of angle measure (which is a non-obvious concern, as I hope I made clear).

Note that with this definition the angle measure belongs to $[0,2\pi]$, with the two extreme points corresponding to different angles (when the two half-lines are the same, and one looks either in between them or outside them) and being difficult to get into the definition (and this way of measuring angles does not exactly fit into the above framework with quotients).

The main reason why I wrote the long answer to end to this relatively easy solution, is that angles are much, much more tricky than any one can realize before trying to write a comprehensive and consistent lecture about this. I had to teach this subject to future high-school teachers, started thinking it would be a piece of cake, then wondered how I could make it fit in my lecture time and student's background without lying to them, and ended lying to them (a bit).

• If the goal is rigor, then I don't see how the definition in terms of arc length helps. Then you just need to define arc length. – Ben Crowell Jan 4 '16 at 14:45
• My goal here is rather getting to a relatively consistent framework which is natural enough to be accepted by the students. In any case, one has to admit a few things, and given that perimeters of rectangles and circles are certainly part of the previous curriculum, arc-length has already been admitted as a notion that can be manipulated and it has a clear meaning. I find more acceptable to use it and wait till later studies to formally define it, rather than e.g. doing the same with $\cos$ and $\sin$ and having to define them analytically rather than geometrically. – Benoît Kloeckner Jan 4 '16 at 16:08
• In any case, one has to admit a few things. Well, if the treatment is based at least loosely on Euclid (which I think is the case for most modern high school texts), then the basic thing you admit is the notion of angle. – Ben Crowell Jan 4 '16 at 16:14

Going all "rigorous definition" in front of students who are being exposed to a concept for the first time in a more formal setting is a serious mistake. All you'll get is glassed over eyes.

You want them to develop an intuition of what angle (or variable, or equation, or polynomial, whatever) is all about, and why the concept is useful. Once that is solid, you can go for rigorous definitions, and show that they agree with the intuitive idea. Once that is internalized, you go for a rigorous axiomatization, and make clear that "angle" is just something unspecified that satisfies the axioms.

That process takes time. Calendar time, not class/study time. Across subjects. Mostly called "developing mathematical maturity".

• Agreed, but the instructor should probably keep the rigorous definition in mind even while initially introducing the intuition to keep any backtracking to minimum when more rigor is eventually required. It's also nice to have up your sleeve for those difficult questions from students that test the teacher! – Dan Christensen Jan 12 '16 at 20:20

None of the proposed definitions that I've seen so far (by snoram, mweiss, and Benoît Kloeckner) actually gives a rigorous definition of the measure of an angle. The define a geometrical figure, but don't define its measure. A clear verbal, conceptual definition is of course helpful to students and needs to be given, but it doesn't constitute a rigorous definition.

One good rigorous approach is to do what Euclid does, but throw out all the bogus pseudo-definitions given in Book I, definitions 1-23. That is, we let notions such as "point," "line," and "angle" be undefined primitive terms. This is the approach used in modern synthetic treatments such as Tarski's, although the number of primitive notions is reduced and usually does not include the measure of an angle. Note that Euclid doesn't verbally distinguish the measure of a figure from the figure itself, but the distinction is always implicitly clear. For example, Book I, Proposition 2 is "To place a straight line equal to a given straight line with one end at a given point." Here he proves that line AL equals line BC, which in more typical modern terminology would be that the measure of line segment AL equals the measure of line segment BC.

For perspective, it may be helpful to consider the question of what is meant by the measure of a line segment. This may seem more basic than the measure of an angle, but logically they have exactly the same status within the axiomatic framework of Euclid. If you want to make one of them a fundamental notion and define the other in terms of it, you can do that.

This is actually a case where I don't think rigor conflicts with education. If you haven't done it, a really interesting experience is to present to a class of students one of Euclid's bogus definitions, and ask them whether they think it's valid and rigorous. When I've done this, the reaction has been that yes, it's rigorous. This shows that, as mweiss has pointed out, "They do not know, yet, what rigor is." After eliciting that reaction, one can explain why modern mathematicians do not accept this kind of definition, and maybe follow up with the famous quote by Hilbert, "One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs."

By the way, one thing that seemed really ugly to me when I looked at the treatment of geometry in my kids' school books was that they assumed two independent mathematical systems, the real numbers and the Euclidean plane. This is completely redundant, and approaching it this way requires lots of "glue" in terms of notation and definitions. The theory of the reals is the theory of Euclidean geometry. They're exactly logically equivalent, and that's why Euclid never had to bother setting up a separate structure for his number system.

• I am sorry but I fail to understand what rigorous definition you propose. Could you extract it from the links? – Benoît Kloeckner Jan 4 '16 at 16:02
• @BenoîtKloeckner: The solution I suggest is to treat "angle" like "point" and "line," as an undefined term. Educationally, I suggest also giving a nonrigorous verbal definition, such as the one proposed by the OP. – Ben Crowell Jan 4 '16 at 16:06
• Ok, sorry, I read a bit too fast. – Benoît Kloeckner Jan 4 '16 at 16:09
• A bit of a problem is that "Euclid's bogus definitions" where considered the utmost in clarity and the ideal to which to strive by untold generations of the most brilliant minds... and you want wet-behind-the-ears high school students to see them as bogus. Won't happen. – vonbrand Jan 11 '16 at 13:01
• Is the measure of an angle really a problem at the high school level? A rigorous definition of any measure may be really complicated. – Dan Christensen Jan 11 '16 at 21:19

Just a suggestion for the purposes of discussion...

Let A, B, C be points in the Euclidean plane.

Then (A,B,C) is said to be an angle with vertex B if and only if B=/=A and B=/=C.

Let A, B, C, D be points in the Euclidean plane such that:

(A,B,C) is an angle with vertex B

D is not on Ray(B,A) or on Ray (B,C)

Then D is said to be an interior point of angle (A,B,C) if and only if there exist points E, F such that:

B=/=E

B=/=F

E is on Ray(B,A)

F is on Ray(B,C)

D is on LineSegment(E,F)