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# Proofs concerning equilateral triangles

Sal proves that the angles of an equilateral triangle are all congruent (and therefore they all measure 60°), and conversely, that triangles with all congruent angles are equilateral. Created by Sal Khan.

Video transcript

What we've got over here is a
triangle where all three sides have the same length,
or all three sides are congruent to each other. And a triangle like this
we call equilateral. This is an equilateral triangle. Now what I want to do is prove
that if all three sides are the same, then we know
that all three angles are going to have the same measure. So let's think how
we can do this. Well, first of
all, we could just look at-- we know that
AB is equal to AC. So let's just pretend
that we don't even know that this also
happens to be equal to BC. And we know for
isosceles triangles, if two legs have
the same length, then the base angles
have the same length. So let's write this down. We know that angle ABC is going
to be congruent to angle ACD. So let me write this down. We know angle ABC is
congruent to angle ACD. So maybe this is my
statement right over here. And then we have reason. And the reason here,
and I'll write it in just kind of shorthand, is
that they're base angles of, I guess you could
say an isosceles. Because we know that this
side is equal to that side. And obviously, this
is an equilateral. All of the sides are equal. But the fact that these
two legs are equal so that the base angles are equal. So we say two legs equal
imply base angles are going to be equal. And that just comes from what we
actually did in the last video with isosceles triangles. But we can also view this
triangle the other way. We could also say that
maybe this angle over here is the vertex angle, and maybe
these two are the base angles. Because then you
have a situation where this side and this side
are congruent to each other. And then that angle
and that angle are going to the base angles. So you could say
angle CAB is going to be congruent to angle ABC,
really for the same reason. We're now looking at different
legs here and different base angles. This would now be the
base in this example. You can imagine turning an
isosceles triangle on its side. But it's the exact same logic. So let's just review
what I talked about. These two sides are
equal, which imply these two base angles are equal. These two sides
being equal implied these two base angles are equal. Well, if ABC is congruent to
ACD and is congruent to CAB, then all of these angles
are congruent to each other. So then we get angle ABC is
congruent to angle ACB, which is congruent to angle CAB. And that pretty much gives
us all of the angles. So if you have an
equilateral triangle, it's actually an equiangular
triangle as well. All of the angles are
going to be the same. And you actually know
what that measure is. If you have three things
that are the same-- so let's call that x, x,
x-- and they add up to 180, you get x plus x plus x is equal
to 180, or 3x is equal to 180. Divide both sides by 3, you
get x is equal to 60 degrees. So in an equilateral
triangle, not only are they all the same
angles, but they're all equal to exactly-- they're
all 60 degree angles. Now let's think about
it the other way around. Let's say I have a triangle. Let's say we've got
ourselves a triangle where all of the angles are the same. So let's say that's point
X, point Y, and point Z. And we know that all
the angles are the same. So we know that this angle
is congruent to this angle is congruent to that angle. So what we showed in the last
video on isosceles triangles is that if two base
angles are the same, then the corresponding legs
are also going to be the same. So we know, for example,
that YX is congruent to YZ. And we know that because the
base angles are congruent. Now we also know that
YZ-- so I'll rewrite YZ-- is congruent to XZ,
by the same argument. But here we're dealing
with different base angles. So now, once again,
you can view this as almost an isosceles
triangle turned on its side. This is the vertex
angle right over here. These are the two base angles. This would be the base now. And we know that because these
two base angles are congruent. So by the same logic. Over in this first
case, the base angles were this angle and that angle. In the second case,
the base angles are that angle and that angle. And actually let
me write it down. The base angles in
this first case-- let me do that
same magenta-- are angle YXZ is congruent
to angle YZX. That was in the first case. These are the base angles. So based on the proof we
saw in the last video, that implies these
sides are congruent. Here, we have these
two base angles. Let me do that in green. Angle XYZ is congruent
to angle YXZ. And so that implies that
these two guys right over here are congruent. Well, there we've proved it. We've said that this side
YX is congruent to YZ. And we've shown that
YZ is congruent to XZ. So all of the sides are
congruent to each other. So once again, if you
have all the angles equal, and they're going to
have to be 60 degrees, then you know that
all of the sides are going to be equal as well. They're going to be congruent.