Degrees in Complementary and Supplementary Angles
Date: 10/28/2002 at 19:18:09 From: Kimberly Subject: Finding degrees in complements and suppliments Hello, I'm in algebra I in the 8th grade. In class we're studying complements and supplements in angles. I do not understand any of the terminology behind the problems. Today we took a test and one of the questions was to find the complement of this angle c degrees, and I didn't even know where to begin. Another was to find the degrees in the third angle in an isosceles triangle, x degrees, x-10 or something like that. Can you explain this a little better? Thanks.
Date: 10/29/2002 at 17:15:37 From: Doctor Ian Subject: Re: Finding degrees in complements and suppliments Hi Kimberly, Part of the problem here is that the names 'complement' and 'supplement' are kind of confusing, since you could assign them the other way around and it would make just as much sense. See: Complementary and Supplementary Angles http://mathforum.org/library/drmath/view/55045.html So if you can't remember which one is to get to 90 degrees and which one is to get to 180 degrees, there's no way to figure it out from first principles. That being the case, how can you keep them straight? The person who runs the Math Forum's Geometry Problem of the Week tells me that she remembers them this way: 'c' comes before 's', and 90 comes before 180. It's the best idea I've heard so far. But terminology aside, what _are_ complements and supplements? If you place two angles next to each other so that they add up to 90 degrees, we say that the angles are complements. | | a / | / a and b are complements | / b |/______ If you place two angles next to each other so that they add up to 180 degrees, we say that the angles are supplements. / / a and b are supplements a / b _____/______ Some examples of complements and supplements are Complements Supplements ------------------ ------------------- 30 and 60 degrees 30 and 150 degrees 2 and 88 degrees 2 and 178 degrees 14 and 76 degrees 14 and 166 degrees Now, note that if you know that two angles are complements or supplements, you can figure out one given the other. How? Well, if they're supplements, you know that they have to add up to 180, this + that = 180 so it must be true that this = 180 - that and that = 180 - this And you can do the corresponding trick for complements, using 90 degrees instead of 180. So whenever you see the phrase 'the supplement of (some angle)', you can immediately translate it to '(180 degrees - (the angle))'. When you have a value for the angle, you end up with something like the supplement of 26 degrees -> (180 - 26) degrees which you can just simplify to get a single number. But if you only have a variable, or an expression for the angle, then you just have to deal with that: the supplement of x-10 degrees -> (180 - (x - 10)) degrees Note that you have to put the expression in parentheses, or you can end up with the wrong thing. In this case, (180 - (x - 10)) is NOT the same as (180 - x - 10) (180 - x + 10) (180 - 10 - x) (190 - x) (170 - x) Why should you care? Well, in geometry, you're constantly dividing things up into triangles in order to make them easier to work with. And in _every_ triangle, the measures of the interior angles add up to 180 degrees. So if you know two angles, the third is the supplement of the sum of the other two. And the nicest kind of triangle to work with is a right triangle. In a right triangle, you have one right angle and two other angles. Since they all have to add up to 180 degrees, and since the right angle takes up 90 of those degrees, the other two angles must add up to 90 degrees. So if you know one of the acute angles in a right triangle, the other is just the complement of that one. So this explains why you need to know about the concepts 'supplement' and 'complement'. But why the jargon? What purpose does that serve? As a rule, people invent jargon because it allows them to say certain things more briefly than they could without the jargon. It's much nicer to be able to say complements x and y instead of angles x and y, whose sum is 90 degrees or the supplement of x instead of the angle you get when you subtract x from 180 degrees And while _you_ may be perfectly happy to use the longer versions, other people will insist on using the shorter ones, and it's important for you to be able to know what they're talking about when they do. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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