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## High school geometry

### Unit 3: Lesson 5

Working with triangles# Finding angles in isosceles triangles

CCSS.Math:

The measures of two angles of an isosceles triangle are 3x+5 and x+16. Find all possible values of x. Created by Sal Khan.

## Video transcript

The measure of two angles
of an isosceles triangle are 3x plus 5 degrees, we'll
say, and x plus 16 degrees. Find all possible values of x. So let's think about this. Let's draw ourselves an
isosceles triangle or two. So it's an isosceles triangle,
like that and like that. And actually, let me
draw a couple of them just because we want
to think about all of the different
possibilities here. So we know, from what we know
about isosceles triangles, that the base angles are
going to be congruent. So that angle is going to
be equal to that angle. That angle is going to
be equal to that angle. And so what could the 3x plus
5 degrees and the x plus 16, what could they be measures of? Well, maybe this
one right over here has a measure of
3x plus 5 degrees. And the vertex is the other one. So maybe this one up here
is the x plus 16 degrees. The other possibility
is that this is describing both base
angles, in which case, they would be equal. So maybe this one is 3x plus
5, and maybe this one over here is x plus 16. And then the final
possibility-- actually we haven't exhausted
all of them-- is if we swap these two--
if this one is x plus 16, and that one is 3x plus 5. So let me draw ourselves
another triangle. And obviously swapping
these two aren't going to make a
difference because they are equal to each other. And then we could make that
one equal to 3x plus 5. But that's not going to
change anything either because they're
equal to each other. So the last situation is
where this angle down here is x plus 16, and this
angle up here is 3x plus 5. This is 3x plus 5. So let's just work
through each of these. So in this situation, if
this base angle is 3x plus 5, so is this base angle. And then we know that
all three of these are going to have to
add up to 180 degrees. So we get 3x plus
5 plus 3x plus 5 plus x plus 16 is going to
be equal to 180 degrees. We have 3x. Let's just add up. You have 3x plus
3x, which gives you 6x, plus another x gives you 7x. And then you have 5
plus 5, which is 10, plus 16 is equal to 26. And that is going
to be equal to 180. And then we have, let's
see, 180 minus 26. If we subtract 26 from both
sides, we get 180 minus 20 is 160, minus another 6 is 154. You have 7x is equal to 154. And let's see how many times--
if we divide both sides by 7, 7 will go into 140 20 times,
and then you have another 14. So it looks like it's 22 times. So x is equal to 22. Is that right? 20 times 7 is 140. 140 plus 14 is 154. So we have x is equal
to 22 degrees in this the first scenario. Now let's think about the
second scenario over here. Now we have these
two characters are going to be equal to each
other because they're both the base angles. So you have 3x plus 5
is equal to x plus 16. Well, you can subtract
x from both sides. And so this becomes 2x
plus 5 is equal to 16. We can subtract
5 from both sides and you get 2x is equal to 11. And then you can
divide both sides by 2, and you get x is equal to 11/2. So that is our second scenario. And then we do our third
scenario right over here. If this base angle is x plus
16, then this base angle right over here is also
going to be x plus 16. They are congruent. And then we can
do the same thing that we did for
the first scenario. All of these angles are going to
have to add up to 180 degrees. So we have x plus 16 plus
x plus 16 plus 3x plus 5. When you add them
all together, you're going to get 180 degrees. Now let's add up
all the x terms. x plus x is 2x plus 3x is 5x. So we get 5x. And then you have
16 plus 16 is 32. 32 plus 5 is 37. Plus 37 is equal to 180 degrees. Subtract 37 from
both sides, and we get 5x is equal to
180 minus 30 is 150. So that gets us to 143. So it's not going
to divide nicely. Divide both sides by 5,
you get x is equal 143/5, which we can just leave
as an improper fraction. You could write it
as a mixed number or however else you
might want to write it. And we're done. These are the three
possible values of x, given the information that
they gave us right up there.