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/