You can’t assume something to be true just because it’s not explicitly stated that it’s not. Think about how convoluted a definition would have to be if that were the case.
Exactly what I wanted to say.
It’s really not hard to do, and is commonly done in mathematics. For example, in this case “Angles are supplementary if and only if they are a pair of angles whose degree measure sums to 180” would suffice. All you require of the definition is both necessary and sufficient conditions (which “if and only if” covers). The problem is that definition 1 in the OP states a sufficient condition, but not a necessary one.
Sure, but all the OP has done is noticed that one of two common definitions is not particularly clear, however the other definition IS clear and a quick browse of the net confirms that both definitions are correct with the second one being more complete. If supplementary applied to more than two angles you would expect to see the first definition appear with other numbers of angles, but it doesn’t.
The problem the OP seems to have is that they think the two definitions somehow contradict each other but they don’t.
From the Oxford English Dictionary (the real one, not an abridged one): supplemental angle, either (in relation to the other) of two angles which are together equal to two right angles.
Which is interesting in a number of ways. One, it is a symmetric definition, and thus really requires that it is limited to two angles. Secondly, it isn’t 180 degress, but two right angles. Which won’t make any difference for Euclidian geometries, but which may matter when a non-Euclidian geometry is considered (maybe).
Of course the main use of the term is as shorthand when describing a proof. You can just say “the supplemantary angle is…” and then define some property of the supplementary angle derived from the manner in which it is created. Just measuring an angle and getting the value of the supplementary is not really what it is for.
Equally, you can’t assume something is false unless it’s explicitly stated. Hence dropping the question if the definition given was ambiguous.
The OP really needs to look at the definition that was used in the class, to see if it specifies exactly two angles.
Not at all. I think one of the definitions is a subset of the other, and I suspect that the narrower one is not universal. There is no value (that I can see) in the narrower definition, and considerable value in the more general one. I’d prefer that my students understand *supplemental *to be that which completes.
Why do you suspect it is not universal? Find an example of supplementary being used to describe more than two angles.
In that case you’re asking for the supplement of 110 degrees which is angle C.
I think it is telling that there are zero google hits for “three angles are supplementary if they add up to 180 degrees.” If your premise was correct then you’d expect a raft of such definitions for supplementary using different numbers of angles, but there aren’t. There aren’t just a few, there are none. Supplementary clearly refers to a pair of angles as described by the more strict definition. Ignoring the strict definition in favour of the ambiguous one does not make your interpretation of the ambiguous one correct.
Version 1 is poorly written and should be modified to say two angles are supplementary. However, it does not look like a definition in the first place. It looks like a statement from a book that was misinterpreted as a definition and may have been copied as though it was one. However, if it was not written as a definition, then it is a true statement. Semantics is very important in Geometry. We and our students must be very specific in our usage. Learning to be accurate, clear and concise in writing is a valuable part of a well designed geometry course.
The statement “Two angles whose measures add to 180 are supplementary” definitely does not preclude three or more angles being supplementary. The second definition is consistent with all Euclidean Geometry I have studied and taught in the past 20 years. I have never seen any text nor, until very recently, heard any teacher suggest the possibility of three angles being supplementary.
If the concept of supplementary angles was taught correctly, there is nothing ambiguous about the question and it should have been kept.
But can a zombie be supplementary?
sunacres, what did you end up doing?