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## Congruent triangles

# Triangle congruence postulates/criteria

CCSS.Math:

## Video transcript

We now know that if
we have two triangles and all of their
corresponding sides are the same, so by
side, side, side-- so if the corresponding
sides, all three of the corresponding sides,
have the same length, we know that those
triangles are congruent. What I want to do
in this video is explore if there
are other properties that we can find between
the triangles that can help us feel pretty good
that those two triangles would be congruent. So side, side, side works. What about angle, angle, angle? So let me do that over here. What about angle angle angle? So what I'm saying
is, is if-- let's say I have a triangle
like this, like I have a triangle like that, and
I have a triangle like this. And if we know that this angle
is congruent to that angle, if this angle is congruent
to that angle, which means that their measures are
equal, or-- and-- I should say and-- and that angle is
congruent to that angle, can we say that these are
two congruent triangles? And at first case, it
looks like maybe it is, at least the
way I drew it here. But when you think
about it, you can have the exact same
corresponding angles, having the same measure
or being congruent, but you could actually scale one
of these triangles up and down and still have that property. For example, if I had this
triangle right over here, it looks similar--
and I'm using that in just the everyday
language sense-- it has the same shape as these
triangles right over here. And it has the same angles. That angle is congruent
to that angle, this angle down here is
congruent to this angle over here, and this
angle over here is congruent to this
angle over here. So all of the angles in all
three of these triangles are the same. The corresponding angles
have the same measure. But clearly, clearly this
triangle right over here is not the same. It is not congruent
to the other two. The sides have a very
different length. This side is much shorter than
this side right over here. This side is much shorter
than that side over there. And this side is much
shorter over here. So with just angle,
angle, angle, you cannot say that a triangle
has the same size and shape. It does have the same shape
but not the same size. So this does not
imply congruency. So angle, angle, angle
does not imply congruency. What it does imply, and we
haven't talked about this yet, is that these are
similar triangles. So angle, angle,
angle implies similar. So let me write it over here. It implies similar triangles. And similar-- you
probably are use to the word in just everyday
language-- but similar has a very specific
meaning in geometry. And similar things have
the same shape but not necessarily the same size. So anything that is
congruent, because it has the same size and
shape, is also similar. But not everything that is
similar is also congruent. So for example, this
triangle is similar-- all of these triangles are
similar to each other, but they aren't all congruent. These two are congruent
if their sides are the same-- I didn't
make that assumption. But if we know that
their sides are the same, then we can say that
they're congruent. But neither of these are
congruent to this one right over here, because this
is clearly much larger. It has the same shape
but a different size. So we can't have an AAA
postulate or an AAA axiom to get to congruency. What about side, angle, side? So let's try this out,
side, angle, side. So let's start off with one
triangle right over here. So let's start off with a
triangle that looks like this. I have my blue side,
I have my pink side, and I have my magenta side. And let's say that I have
another triangle that has this blue side. It has the same side, same
length as that blue side. So let me draw it like that. It has the same length
as that blue side. So that length and that length
are going to be the same. It has a congruent
angle right after that. So this angle and the next
angle for this triangle are going to have
the same measure, or they're going
to be congruent. And then the next
side is going to have the same length as
this one over here. So that's going to be the
same length as this over here. So it's going to
be the same length. And because we only know that
two of the corresponding sides have the same length, and
the angle between them-- and this is important--
the angle between the two corresponding sides also
have the same measure, we can do anything we want with
this last side on this one. We can essentially--
it's going to have to start right over here. You could start from this point. And we can pivot it to
form any triangle we want. But we can see, the only
way we can form a triangle is if we bring this side
all the way over here and close this right over there. And so we can see just
logically for two triangles, they have one side that
has the length the same, the next side has
a length the same, and the angle in between
them-- so this angle-- let me do that in the
same color-- this angle in between them, this is the angle. This A is this angle
and that angle. It's the angle in between them. This first side is in blue. And this second side right,
over here, is in pink. Well, it's already
written in pink. So we can see that if two
sides are the same, have the same length-- two
corresponding sides have the same length, and
the corresponding angle between them, they
have to be congruent. There's no other one place
to put this third side. So SAS-- and sometimes,
it's once again called a postulate, an axiom,
or if it's kind of proven, sometimes is called
a theorem-- this does imply that the two
triangles are congruent. So we will give ourselves
this tool in our tool kit. We had the SSS postulate. Now we have the SAS postulate. Two sides are equal and
the angle in between them, for two triangles,
corresponding sides and angles, then we can say that
it is definitely-- these are congruent triangles. Now what about--
and I'm just going to try to go through all the
different combinations here-- what if I have
angle, side, angle? So let me try that. So what happens if I
have angle, side, angle? So let's go back to this
one right over here. So actually, let me
just redraw a new one for each of these cases. So angle, side, angle, so
I'll draw a triangle here. So I have this triangle. So this would be maybe the side. That would be the side. So let me draw the whole
triangle, actually, first. So I have this triangle. Let me draw one side over here. And then let me draw
one side over there. And this angle right
over here, I'll call it-- I'll do it in orange. And this angle over here,
I will do it in yellow. So if I have another
triangle that has one side having
equal measure-- so I'll use it as this
blue side right over here. So it has one side
that has equal measure. And the two angles on
either side of that side, or at either end of
that side, are the same, will this triangle
necessarily be congruent? And we're just going to
try to reason it out. These aren't formal proofs. We're really just
trying to set up what are reasonable
postulates, or what are reasonable assumptions
we can have in our tool kit as we try to prove other things. So that angle, let's call it
that angle, right over there, they're going to have the
same measure in this triangle. And this angle right
over here in yellow is going to have the same
measure on this triangle right over here. So regardless,
I'm not in any way constraining the
sides over here. So this side right over
here could have any length. It could have any length, but it
has to form this angle with it. So it could have any length. And it can just go as
far as it wants to go. In no way have we constrained
what the length of that is. And actually, let me
mark this off, too. So this is the same as this. So that side can be anything. We haven't constrained it all. And once again, this
side could be anything. We haven't
constrained it at all. But we know it has
to go at this angle. So it has to go at that angle. Well, once again,
there's only one triangle that can be formed this way. We can say all day
that this length could be as long as we want
or as short as we want. And this one could be as
long as we want and as short as we want. But the only way that they
can actually touch each other and form a triangle and
have these two angles, is if they are the exact same
length as these two sides right over here. So this side will actually have
to be the same as that side. And this would have to
be the same as that side. Once again, this isn't a proof. I'd call it more of a reasoning
through it or an investigation, really just to establish
what reasonable baselines, or axioms, or assumptions, or
postulates that we could have. So for my purposes, I
think ASA does show us that two triangles
are congruent. Now let's try another one. Let's try angle, angle, side. Let's try angle, angle, side. And in some geometry
classes, maybe if you have to go
through an exam quickly, you might memorize, OK, side,
side, side implies congruency. And that's kind of logical. Side, angle, side
implies congruency, and so on, and so forth. I'm not a fan of memorizing it. It might be good
for time pressure. It is good to, sometimes, even
just go through this logic. If you're like, wait, does
angle, angle, angle work? Well, no, I can
find this case that breaks down angle, angle, angle. If these work, just try
to verify for yourself that they make logical sense
why they would imply congruency. Now, let's try
angle, angle, side. Let's try angle, angle, side. So once again, let's have
a triangle over here. It has some side. So this one is going to be a
little bit more interesting. So it has some side. That's the side
right over there. And then, it has two angles. So let me draw the other
sides of this triangle. I'll draw one in magenta
and then one in green. And there's two angles
and then the side. So let's say you
have this angle-- you have that angle
right over there. Actually, I didn't
have to put a double, because that's the first angle
that I'm-- So I have that angle, which we'll refer
to as that first A. Then we have this angle,
which is that second A. So if I know that there's
another triangle that has one side having the
same length-- so let me draw it like that-- it has one
side having the same length. It has one angle on that side
that has the same measure. So it has a measure like that. And so this side right over
here could be of any length. We aren't constraining what
the length of that side is. But whatever the angle is on
the other side of that side is going to be the same as this
green angle right over here. So for example, it
could be like that. And then you could have a
green side go like that. It could be like that and have
the green side go like that. And if we have--
so the only thing we're assuming is that this
is the same length as this, and that this angle is the
same measure as that angle, and that this measure is the
same measure as that angle. And this magenta line
can be of any length, and this green line
can be of any length. We in no way have
constrained that. But can we form any triangle
that is not congruent to this? Because the bottom line
is, this green line is going to touch this
one right over there. And the only way it's going to
touch that one right over there is if it starts right
over here, because we're constraining this
angle right over here. We're constraining that angle. And so it looks
like angle, angle, side does indeed
imply congruency. So that does imply congruency. So let's just do one more
just to kind of try out all of the different situations. What if we have--
and I'm running out of a little
bit of real estate right over here at
the bottom-- what if we tried out
side, side, angle? So once again, draw a triangle. So it has one side there. It has another side there. And then-- I don't have to
do those hash marks just yet. So one side, then another
side, and then another side. And what happens if we know that
there's another triangle that has two of the sides the same
and then the angle after it? So for example, we would have
that side just like that, and then it has another side. But we're not
constraining the angle. We aren't constraining
this angle right over here, but we're constraining
the length of that side. So let me color code it. So that blue side
is that first side. Then we have this magenta
side right over there. So this is going to be the same
length as this right over here. But let me make it
at a different angle to see if I can disprove it. So let's say it looks like that. Or actually let me make
it even more interesting. Let me try to make it like that. So it's a very different angle. But now, it has to have
the same angle out here. It has to have that
same angle out here. So it has to be
roughly that angle. So it actually looks like we
can draw a triangle that is not congruent that has two
sides being the same length and then an angle is different. For example, this
is pretty much that. I made this angle
smaller than this angle. These two sides are the same. This angle is the same now, but
what the byproduct of that is, is that this green side is going
to be shorter on this triangle right over here. So you don't necessarily
have congruent triangles with side, side, angle. So this is not
necessarily congruent, not necessarily, or similar. It gives us neither
congruency nor similarity.