# Basic geometry definition: Can three angles be supplementary?

I find that the definition of supplementary angles is given in a couple of different ways.

1. Two Angles are Supplementary if they add up to 180 degrees.

or

1. Supplementary angles are pairs of angles whose measures add up to 180 degrees.

When the measures of 3 angles add up to 180 degrees, version 2 says they are not supplementary, but version 1 does not exclude that possibility, therefore seems to allow it.

Which is “correct”?

Huh? Your first definition clearly says that it’s two angles.

I don’t think it does. The second one does, for sure (“supplementary angles are pairs”), but the first seems to says that “pairs are supplementary angles,” not “only pairs are supplementary.”

I’m going to have to agree with Snarky Kong. “1. Two Angles are Supplementary if they add up to 180 degrees” says “two angles,” not three.

Further investigation suggests that version 2 above (from wikipedia, incidentally) is in subtle conflict with The American Heritage Dictionary of the English Language, which doesn’t define supplementary, but says under supplement:

1. Mathematics The angle or arc that when added to a given angle or arc makes 180° or a semicircle. Also called supplementary angle.

This definition seems to say that only one angle is supplementary, its always supplementary to another “existing” arc or angle. According to this definition (which seems pretty reasonable to me), even if there are a dozen angles which sum to 180 degrees, each is supplementary to the rest.

Is that “correct”?

If you need to talk about three or more angles which sum to 180 degrees, and you want to call them supplementary, I don’t think people would object, since it’s a natural extension of the definition.

It sounds like you’re more worried about the semantics of it, though.

If you’re going to go this route, explain that you’re using the terms this way. If you just drop such a use in the middle of a sentence without forewarning, you’ll end up annoying your readers.

That’s my conclusion too.

The context is this: I’m teaching a section of High School Geometry. The “lead teacher” (not me) writes the tests and yesterday I gave a test to my students that I hadn’t seen prior to class. I’m just now sitting down to grade them. One of the questions was:

Three angles can be supplementary. T F

The lead teacher’s answer key says False, but it looks to me to be a poorly conceived question.

Wikipedia says that supplementarity is a property of pairs of angles, and based on common usage, that’s correct. So I agree with the answer key, but I also agree that some of the definitions given can be ambiguous.

I agree that Wikipedia is describing common usage, but according to one of its citations (Websters) it seems to be adding content in this case.

Our Geometry textbook (Larson, Boswell & Stiff from McDougall Littell, pretty widely used) uses the language in 1. of the OP, not the Wikipedia one. The case of whether three angles summing to 180 can properly be called “supplementary” hasn’t come up in class, but according to every dictionary definition I’ve found so far it “is not incorrect.”

I would have been quite content to nail students on this if I’d made a point of it in class, but I didn’t and after surveying the references I’m inclined to toss out the question. The idea of restricting the definition to pairs of angles seems oversimplified.

Just because something is not specifically excluded does not mean it is allowed. “Supplementary” is a property that two angles can have, it is not a property that any other number of angles can have.

Both of your definitions are correct, number one may be open to interpretation, however number two clarifies.

http://www.algebralab.org/lessons/lesson.aspx?file=Geometry_AnglesComplementarySupplementaryVertical.xml

That’s what your cite says, but as noted above not every authority agrees. I’m inclined to suspect that the wording “Supplementary angles are two angles with a sum of 180º” is a simplification, not a fully (though perhaps widely) accepted one.

Anyway, my problem of what to do with the tests is solved (omit the problem from the scoring) but it remains for me to have this conversation with the department to see if we need to take a stand one way or the other. Thanks all.

I think the point of “supplementary” is to say “opposite.” As in, “I have an angle of 15 degrees. What is its opposite? Yup, 165.” You have 1 angle and 1 supplement.

Granted, the teacher’s question was dumb, testing a completely superfluous point, and liable to be incorrectly answered by smart kids, which is unfair. Omitting the question from grading was a very good move on your part.

I agree it is a bad question and from a teaching point of view I think you should exclude it. Smart kids could have thought the answer is false for good reasons. It just seems like a filler question anyway - what knowledge or skill is it testing for?

By the way, I feel your pain. The language and vocabulary of maths is a tricky issue in primary and secondary education and some teachers deal with it badly or don’t think it is important at all.

Scratch that, the point of “supplementary” is to say, “the other thing this angle might be called.” Consider it from an engineering, rather than a math, perspective. As in, “So, you want this piece of wood cut at 60 degrees?” “No, 120.” “Do you want it pointy or not?” “Yes!” “So why did you say 120? 60 is pointy!” “I was telling you the angle to set on the saw!” And so on…

Frankly, I’m starting to wonder why teach this word to kids at all.

So you’ve omitted the question, but how did the students answered the question? I’m guessing most of the answers were “False”, so the students were probably not as confused by the question as you were.

Can you find any cite that explicitly supports more than two angles being supplementary? I say that every authority does agree but some use the slightly more sloppy wording. Even the ones that talk about definition 1 still make no mention of more than two angles being supplementary, so even though their definition apparently allows for it, they are still using it in the same sense as definition 2.

The dictionary discussion suggests that the point of “supplementary” is to say “the stuff you have to add in order to make something whole.” In the case of more than two angles one could decide that you have to sum them in order to calculate the supplement, so in the end it’ll be two being compared anyway.

However, I think it is valid to ask “angle A and angle B are adjacent and their measures sum to 110 degrees, what is their supplement?”

whoops :smack: it’s actually used extensively in classical geometry, geometric proofs, construction, etc. But that brings up a point… does it make any sense to call three angles supplementary in the context of those?

Nope, no counterexamples at hand.

In my textbook’s examples of three non-parallel, coplanar lines intersecting at one point, they point out that “any three consecutive adjacent angles sum to 180 degrees” without saying that “they are supplementary.”

But they never say that they aren’t!