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# Congruent triangles & the SSS postulate/criterion

CCSS Math: HSG.CO.B.7

## Video transcript

Let's talk a little
bit about congruence. And one way to think
about congruence, it's really a kind of
equivalence for shape. So in algebra when something
is equal to another thing, it means that their
quantities are the same. But if we're now all of a
sudden talking about shapes, and we say that those shapes
are the same size and shape, then we say that
they're congruent. And just to see a
simple example here. I have this triangle
right over there. And let's say I have this
triangle right over here. And if you are able
to shift this triangle and rotate this triangle
and flip this triangle, you can make it look
exactly like this triangle. As long as you're not
changing the lengths of any of the sides
or the angles here. But you can flip it, you
can shift it, and rotate it. Let me write this. You can shift it, you can
flip it, and you can rotate. If you can do those
three procedures to make these the exact same triangle,
to make them look exactly the same, then
they are congruent. And if you say that a
triangle is congruent-- let me label these. So let's call this
triangle A, B, and C. And let's call this
X, Y, and Z. So if we were to say,
if we make the claim, that both of these
triangles are congruent, so if we say triangle
ABC is congruent-- and the way you specify
it, it looks almost like an equal sign,
but it's an equal sign with this little
curly thing on top. Let me write a
little bit neater. So we would write it like this. If we know that triangle ABC
is congruent to triangle XYZ, that means that their
corresponding sides have the same length, and
their corresponding angles have the same measure. So if we make this assumption,
or if someone tells us that this is true, then
we know, for example, that AB is going
to be equal to XY. The length of
segment AB is going to be equal to the
length of segment XY. And we could denote
it like this. And I'm assuming that these
are the corresponding sides. And you can see it,
actually, by the way we've defined these triangles. A corresponds to X,
B corresponds to Y, and then C corresponds
to Z right over there. So side AB is going to have
the same length as side XY. And you can sometimes, if
you don't have the colors, you would denote
it just like that. These two line segments
have the same length. And you could actually say
this, and you don't always see it written this way, you
could also make the statement that line segment AB is
congruent to line segment XY. But congruence of
line segments really just means that their
lengths are equivalent. So these two things
mean the same thing. If one line segment is congruent
to another line segment, that just means the measure
of one line segment is equal to the measure
of the other line segment. And so we can go through
all the corresponding sides if these two characters
are congruent. We also know the length of BC
is going to be the length of YZ, assuming that those are
the corresponding sides. And we could put these double
hash marks right over here to show that these two
lengths are the same. And then if we go
to the third side, we also know that these are
going to have the same length, or the line segments themselves
are going to be congruent. So we also know that
the length of AC is going to be equal
to the length of XZ. Not only do we know that all the
corresponding sides are going to have the same length,
if someone tells us that a triangle is
congruent, we also know that all the
corresponding angles are going to have
the same measure. So for example, we also know
that this angle's measure is going to be the same as the
corresponding angle's measure. The corresponding angle
is right over here. It's between this orange
side and this blue side. Or this orange side and this
purple side, I should say. In between this orange
side and this purple side. And so it also tells us that
the measure of angle BAC is equal to the
measure of angle YXZ. Let me write that angle
symbol a little less like a-- We could also write
that, as angle BAC is congruent to angle YXZ. And once again,
like line segments, if one line segment is congruent
to another line segment, it just means that
their lengths are equal. And if one angle is
congruent to another angle, it just means that their
measures are equal. So we know that those two
corresponding angles have the same measure,
they're congruent. We also know that these two
corresponding angles have the same measure, and
I'll use a double arc to specify that this has
the same measure as that. So we also know that
the measure of angle ABC is equal to the
measure of angle XYZ. And then finally, we
know that this angle, if we know that these two
characters are congruent, that this angle is going to have
the same measure as this angle, as its corresponding angle. So we know that the
measure of angle ACB is going to be equal to
the measure of angle XZY. Now what we're going to concern
ourselves a lot with, is how do we prove congruence? Because it's cool, because
if you can prove congruence of two triangles,
then all of a sudden you can make all of
these assumptions. And what we're going to find
out, and this is going to be, we're going to assume
it for the sake of kind of an introductory
geometry course that this is an axiom,
or postulate, or just something that you assume. And let me actually
just write this down. So an axiom, very fancy word. Postulate, also a
very fancy word. It really just means things that
we're going to assume are true. An axiom is sometimes, there's
a little bit of distinction sometimes, where someone will
say an axiom is something that's self-evident, or it
seems like a universal truth that is definitely true. And we just take it for granted. You can't prove an axiom. A postulate kind of
has that same role, but sometimes you
could say, well, let's just assume
this is true and see, if we assume that that's true,
what we can derive from it. What we can prove if
we assume it's true. But for the sake of a
introductory geometry class, and really in most
of mathematics today, these two words
are used interchangeably. An axiom or postulate,
just very fancy words for things that we
take as a given. Things that we just will assume. We won't prove them. We will start with
these assumptions, and then we're just going
to build up from there. And one of the core ones
that we'll see in geometry is the axiom, or the postulate,
that if all the sides are congruent, or if the lengths of
all the sides of the triangle are congruent,
then we are dealing with congruent triangles. So it's sometimes called
a side-side-side postulate or axiom. We're not going
to prove it here. We're just going to
take it as a given. So this literally stands
for side-side-side. And what it tells us is, if
we have two triangles-- and so let's say that's, and
that's another triangle right over there-- and we
know that corresponding sides are equal. So we know that this side right
over here is equal in length to that side, we know that
this side right over here is equal in length to that
side right over there, and we know that this side
over here is equal in length to this side over here. Then we know, and
we're just going to take this as an assumption,
and can build up off of this, we know that they are congruent. That these two triangles
are congruent to each other. I didn't put any labels there,
so it's kind of hard for me to refer to them. But these two are
congruent triangles. What's powerful
there is if we know all the corresponding
sides are equal, then we know they're
congruent, and then we can make all the other
assumptions, which means that the corresponding
angles are also equal. So that we know that that's
going to be congruent to that, or have the same measure. That's going to have the
same measure as that. And then that is going
to have the same measure as that right over there. And to see why that
is a reasonable axiom, or a reasonable assumption,
or a reasonable postulate to start off with, let's
start with one triangle. So let's say I have this
triangle right over here. So it has that side, and
then it has that side, and then it has that
side right over here. And what I'm going to do is see,
if I have another triangle that has the exact same side
lengths, is there any way for me to construct a triangle with
the same side lengths that is different, that can't be
translated to this triangle through flipping,
shifting, or rotating. So we assume this
other triangle's going to have a side that's
the same length as that one right over there. So I'll try to
draw it like that. Roughly the same length. We know it's going to have
a side that's that length. Let me put on this
side just to make it look a little bit
more interesting. So we know it's going to
have a side like that. So I'm going to draw it
roughly the same length, but I'm going to try to
do it a different angle. Now we know it's going to have
a side that looks like that. And so let me, I'll
put it right over here. It's about that length
right over there. And so clearly, this
isn't a triangle. In order to make
it a triangle, I'll have to connect this point to
that point right over there. And really there's
only two ways to do it. I can rotate it around that
little hinge right over there. If I connect them
over here, then I'm going to get a triangle
that looks like this, which is really just a flipped--
am I visualizing it right? Yeah, just a flipped version. You can rotate it a
little bit back this way. But then you'd have the
magenta on this side and the yellow on this side. And you could flip
it vertically, and then it would look
exactly like this. Our other option to make
these two points connect, is to rotate them out this way. And then the yellow side
is going to be here, and then the magenta side is
going to be right over here. That's not magenta. The magenta side is going
to be just like that. And if we do that,
then we actually just have to rotate it around
to get that exact triangle. So this isn't a proof,
but actually we're going to start assuming
that this is an axiom. But hopefully you see that
it's a pretty reasonable starting point, that if
all of the sides, all the corresponding sides,
of two different triangles are equal, then we know
that they are congruent. We're just going to assume,
it is an axiom that we're going to build off of,
that they're congruent. And then we also know that
all the corresponding angles are going to be equivalent.