Here we have a popular word problem that talks about complementary angles.
It says uhm, two angles, alpha and beta, alpha and beta are just simply Greek letters, some
books will use a, and b, or x, and y, the letters don't matter.
But uhm, two angles, alpha and beta are complementary.
Now what, what, does complementary mean?
It simply means that these guys add up to ninety degrees.
So sixty and thirty are complementary.
Forty-five and forty-five are complementary.
Eighty and ten are complementary.
Any two angles that add up to ninety would be considered complementary.
It says find alpha and beta if alpha is eighteen degrees less than beta.
So, uhm, one thing we know right off the bat is that alpha plus beta adds up to ninety.
Uhm, but the problem with that is there's two variables here, and we can only solve
this as a linear equation if there's only a single variable.
But notice they give us a little bit of added information in the second line.
They tell us that alpha is eighteen degrees less than beta.
So we can write that as a little equation.
We say alpha, equals� beta, minus eighteen.
Because whatever beta is, if you take eighteen degrees off of that, you'll get alpha, based
off of what it is saying here.
Make sure you realize it's not eighteen minus beta, it's beta minus eighteen, that's important.
So how do we take this main equation and reduce it from two variables, down to just a single
variable?
Well, look at this, they tell us what alpha is.
Alpha and beta minus eighteen are the same thing.
So we can do a cut and paste.
We can take this alpha out, and instead of alpha plus beta equals ninety, we can take
the alpha out, and replace it with beta minus eighteen.
Because that's also the same thing as alpha.
And so yes, this equations longer, but it has a benefit, this yellow equation only has
one variable, it just has beta.
So we can solve it like a regular equation.
We'll combine like terms, I have a beta, plus beta, that makes two beta.
Minus eighteen equals ninety.
And then we'll put the variables on one side, and move the constants to the other.
Betas on the left, so let's take this minus eighteen, and we'll add eighteen to both sides.
Uhm, so that will give us two beta.
The minus eighteen, plus eighteen will cancel, and ninety, plus eighteen, would be a hundred
and eight.
And then divide by the variable's coefficient.
Divide by two, beta would be fifty-four degrees.
Okay now, that's just the beta.
They actually want alpha and beta.
So we have a little bit more work to do.
Uhm, so how do we get the alpha, if what we currently have is only the beta?
Well you can use any old equation to do this.
You could uhm, either use ah, this equation here, alpha plus beta adds up to ninety.
Or, I think I might actually use this equation, alpha equals beta, minus eighteen.
That one's okay as well.
So based off of that, this says that alpha would be fifty-four minus eighteen.
So that means that alpha ah, would be thirty-six.
Thirty-six degrees, in fact.
So here we go, we have our final answer.
Final answer, alpha is thirty-six degrees, while beta is fifty-four degrees.
And it's always a good idea when you, when you, finish, when you're done, just to take
a quick ah, glance over the problem, and see if all of the criteria have been satisfied.
Ah, first thing I'm, I'm thinking about is, are these two really complementary?
Do they add up to ninety?
Thirty-six plus fifty-four does give you ninety.
So it�s because they add up to ninety, these two are complementary.
And the other criteria was the fact that alpha had to be eighteen degrees less than beta.
So what's, what's, the difference between fifty-four, and thirty-six?
Well its eighteen degrees, alpha really is eighteen degrees less than beta.