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## Theorems concerning triangle properties

# Proofs concerning isosceles triangles

CCSS.Math:

## Video transcript

So we're starting off
with triangle ABC here. And we see from the
drawing that we already know that the length of AB
is equal to the length of AC, or line segment AB is
congruent to line segment AC. And since this is a triangle
and two sides of this triangle are congruent, or they
have the same length, we can say that this is
an isosceles triangle. Isosceles triangle, one of the
hardest words for me to spell. I think I got it right. And that just means
that two of the sides are equal to each other. Now what I want to
do in this video is show what I want to prove. So what I want to prove here
is that these two-- and they're sometimes referred to as base
angles, these angles that are between one of the
sides, and the side that isn't necessarily equal to
it, and the other side that is equal and the side
that's not equal to it. I want to show that
they're congruent. So I want to prove
that angle ABC, I want to prove that that
is congruent to angle ACB. And so for an
isosceles triangle, those two angles are
often called base angles. And this might be called
the vertex angle over here. And these are often
called the sides or the legs of the
isosceles triangle. And these are--
obviously they're sides. These are the legs of the
isosceles triangle and this one down here, that
isn't necessarily the same as the other two,
you would call the base. So let's see if
we can prove that. So there's not a
lot of information here, just that these
two sides are equal. But we have, in
our toolkit, a lot that we know about
triangle congruency. So maybe we can construct
two triangles here that are congruent. And then we can use
that information to figure out whether this
angle is congruent to that angle there. And the first step, if
we're going to use triangle congruency, is to actually
construct two triangles. So one way to construct
two triangles is let's set up another point
right over here. Let's set up another
point D. And let's just say that D is the midpoint of
B and C. So it's the midpoint. So the distance
from B to D is going to be the same thing
as the distance-- let me do a double slash
here to show you it's not the same
as that distance. So the distance
from B to D is going to be the same thing as
the distance from D to C. And obviously, between any two
points, you have a midpoint. And so let me draw segment AD. And what's useful about
that is that we have now constructed two triangles. And what's even cooler is that
triangle ABD and triangle ACD, they have this
side is congruent, this side is congruent,
and they actually share this side right over here. So we know that
triangle ABD we know that it is congruent
to triangle ACD. And we know it because
of SSS, side-side-side. You have two triangles that have
three sides that are congruent, or they have the same length. Then the two triangles
are congruent. And what's useful about that
is if these two triangles are congruent, then their
corresponding angles are congruent. And so we've actually
now proved our result. Because the corresponding
angle to ABC in this triangle is angle ACD in this
triangle right over here. So that we then know that angle
ABC is congruent to angle ACB. So that's a pretty neat result. If you have an
isosceles triangle, a triangle where two of
the sides are congruent, then their base angles,
these base angles, are also going to be congruent. Now let's think about
it the other way. Can we make the other statement? If the base angles
are congruent, do we know that these two legs
are going to be congruent? So let's try to
construct a triangle and see if we can
prove it the other way. So I'll do another
triangle right over here. Let me draw another
one just like that. That's not that
pretty of a triangle, so let me draw it
a little nicer. I'm going to draw it like this. Let me do that in
a different color. So I'll call that A.
I will call this B. I will call that C
right over there. And now we're going to
start off with the idea that this angle, angle ABC,
is congruent to angle ACB. So they have the
same exact measure. And what we want to
do in this case-- we want to prove-- so let
me draw a little line here to show that we're
doing a different idea. Here we're saying if these
two sides are the same, then the base angles are
going to be the same. We've proved that. Now let's go the other way. If the base angles
are the same, do we know that the two
sides are the same? So we want to prove that
segment AC is congruent to AB. Or you could say that
the length of segment AC, which we would
denote that way, is equal to the length
of segment AB. These are essentially
equivalent statements. So let's see. Once again in our toolkit, we
have our congruency theorems. But in order to apply
them, you really do need to have two triangles. So let's construct
two triangles here. And this time, instead
of defining another point as the midpoint, I'm going to
define D this time as the point that if I were to
go straight up, the point that is
essentially-- if you view BC as straight horizontal, the
point that goes straight down from A. And the reason
why I say that is there's some point-- you could
call it an altitude-- that intersects BC at a right angle. And there will definitely
be some point like that. And so if it's a right
angle on that side, if that's 90
degrees, then we know that this is 90 degrees as well. Now, what's
interesting about this? And let me write this down. So I've constructed AD such
that AD is perpendicular to BC. And you can always
construct an altitude. Essentially, you just have to
make BC lie flat on the ground. And then you just have
to drop something from A, and that will give you
point D. You can always do that with a
triangle like this. So what does this give us? So over here, we have
an angle, an angle, and then a side in common. And over here, you have an angle
that corresponds to that angle, an angle that corresponds
to this angle, and the same side in common. And so we know that these
triangles are congruent by AAS, angle-angle-side,
which we've shown is a valid congruent postulate. So we can say now
that triangle ABD is congruent to triangle ACD. And we know that by
angle-angle-side. This angle and this
angle and this side. This angle and this
angle and this side. And once we know these two
triangles are congruent, we know that every corresponding
angle or side of the two triangles are also
going to be congruent. So then we know that AB is
a corresponding side to AC. So these two sides
must be congruent. And so you get AB is going
to be congruent to AC, and that's because these
are congruent triangles. And we've proven what
we wanted to show. If the base angles are
equal, then the two legs are going to be equal. If the two legs are equal,
then the base angles are equal. It's a very, very, very
useful tool in geometry. And in case you're curious,
for this specific isosceles triangle, over here we set
up D so it was the midpoint. Over here we set up D so
it was directly below A. We didn't say whether
it was the midpoint. But here, we can actually show
that it is the midpoint just as a little bit of a bonus
result, because we know that since these two
triangles are congruent, BD is going to be
congruent to DC because they are the
corresponding sides. So it actually
turns out that point D for an isosceles triangle,
not only is it the midpoint but it is the place where,
it is the point at which AD-- or we could say that AD is a
perpendicular bisector of BC. So not only is AD perpendicular
to BC, but it bisects it. That D is the midpoint
of that entire base.