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## Triangle congruence from transformations

# Why SSA isn't a congruence postulate/criterion

CCSS.Math:

## Video transcript

Several videos
ago, I very quickly went through why side-side-angle
is not a valid postulate. And what I want to
do in this video is explore it a little bit more. And it's not called
angle-side-side for obvious reasons,
because then the acronym would make people giggle
in geometry class. And I guess we don't
want people giggling while they're doing mathematics. So let's just think
about a triangle here. So let's say I have a triangle. Let me draw it. Let's have a triangle that
looks something like this. If I have a triangle
that looks something-- I have trouble drawing
straight triangles. So let's say the triangle
looks something like that. And let's say that we've
found another triangle that has a congruent
side, a side that is congruent to this
side right over here. I guess any side on a triangle
is next to the other two sides. Next to that is a side that
is congruent to this side right over here. And then that side is one
of the sides of an angle. So it forms one of the parts
of an angle right over here. And that other triangle
has a congruent angle right over here. So this is the angle that that
first side is not a part of. Only that second side
is part of this angle. So this is side-side-angle. Or you could call
it angle-side-side and giggle a little
bit about it. Now, how do we know that
this doesn't by itself show that this is congruent? Well, we'd have to show
that this could actually imply two different triangles. And to think about
that, let's say we know that the angle-- we know
that this other triangle has that same yellow
angle there, which means that the blue side is
going to have to look something like that, just the way
we drew it over here. This side down here, I'll
make it a green side. This green side down here
we know nothing about. We never said that this
is congruent to anything. If we knew, then we
could use side-side-side. We only know that
this side is congruent and this side is congruent,
and this angle is congruent. So this green side, and I'll
draw it as a dotted line, it could be of any length. We don't know what the
length is of that green side. Now we have this magenta side. We have another side
that is congruent here. So this thing could
pivot over here. We know nothing about this angle
so it could form any angle. But it does have to
get to this other side. So one possibility is that maybe
the triangles are congruent. So maybe this side
does go down just like that, in which
case, we actually would have congruent triangles. But the kind of aha
moment here, or the reason why SSA isn't possible, is
that this side, could also come down like this. There's two ways to get
down to this base, if you want to call it that way. It can come out that way or it
could kind of come in this way. And so that's why SSA by itself
with no other information is ambiguous. It does not give you
enough information to say that those triangles
are definitely the same. Now there are special cases. So in this situation
right over here, our angle, the angle in our
SSA, our angle was acute. This is an acute
angle right over here. And when you have an
acute angle as one of the sides of your
triangle, the other sides of the triangle, you could
still have an obtuse angle. Remember, acute
means less than 90 degrees, obtuse means
greater than 90 degrees. So you could still
have an obtuse angle. So that's why this is an option. So one option is that you
have two other acute angles. So this could be acute. This is also acute,
also acute, also acute. But then you have
the option where this is even more
acute, even narrower, and then this becomes
an obtuse angle. And that's only possible if
you don't-- you can't have two obtuse angles in
the same triangle. You can't have two
things that have larger than 90-degree measure
in the same triangle. And so that's why
there is a possibility where if you have another
triangle that looks like this, and if I were to tell you
very clearly that this angle right over here is obtuse--
and that is the A in the SSA. So you have the angle. And I were to say I have another
triangle where this angle is congruent to that other
triangle, some angle of that other triangle, and then
one of the sides adjacent to it is congruent, and then the next
side over is also congruent, then it's not so ambiguous. Because we could
try to draw that. So let's draw that same
congruent obtuse angle. We know nothing about
this side down here because we haven't said that
that's necessarily congruent. So that could be of any length. We do know that this
triangle is going to have the same length
for this side of the angle. So it looks like this. And then we know
that this side-- let me do that in orange. We know that this side is also
going to be the same length. We haven't told you
anything about this angle right over here. So this side could
pivot over here. We can kind of
rotate it over there. But there's only one way,
now, that this orange side can reach this green side. Now the only way is
this way over here. And we were more constrained,
or this case isn't ambiguous, because we used up
our obtuse angle here. The A here is an obtuse one. And so then it constrains
what the triangle can become. So I don't want to make
you say, in general, SSA, you do not want to
use it as a postulate. I just wanted to make
it clear that there is the special case where if
you know that the A in the SSA is obtuse, then it becomes
a little bit less ambiguous. And then finally,
there's a circumstance that this is an acute angle
where it would be ambiguous. You have the obtuse
angle, and then you have something in between,
which is the right angle. So where you have the A
in SSA is a right angle. So if you had it like this. If you have a
right angle and you have some base of unknown length
but you fix this length right over here-- if you know that
this is fixed because you're saying it's congruent
to some other triangle, and if you know that the
next length is fixed-- and if you think about
it, this next side is going to be the side
opposite the right angle. It's going to have to be the
hypotenuse of the right angle. Then you know that the only
way you can construct this, and similar to the
obtuse case, and if you know the length of this,
the only way you could do it is to bring it down over here. So that actually does
lead to another postulate called the right angle side
hypotenuse postulate, which is really just a
special case of SSA where the angle is
actually a right angle. And here, they wrote
the angle first. You could view this
as angle-side-side. And they were able to
do it because now they can write "right angle,"
and so it doesn't form that embarrassing acronym. And this would also be a
little bit common sense. Because if you know two
sides of a right triangle-- and we haven't gone into
depth on this in the geometry playlist, but you might
already be familiar with it-- by Pythagorean
theorem, you can always figure out the third side. So if you have this
information about any triangle, you can always figure
out the third side. And then you can
use side-side-side. So I just wanted to show you
this little special case. But in general,
the important thing is that you can't just
use SSA unless you have more information.