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CCSS.Math:

Let's say we have triangle ABC. It looks something like this. I want to think about the
minimum amount of information. I want to come up with
a couple of postulates that we can use to determine
whether another triangle is similar to triangle ABC. So we already know
that if all three of the corresponding
angles are congruent to the corresponding
angles on ABC, then we know that we're dealing
with congruent triangles. So for example, if
this is 30 degrees, this angle is 90 degrees, and
this angle right over here is 60 degrees. And we have another
triangle that looks like this, it's
clearly a smaller triangle, but it's corresponding angles. So this is 30 degrees. This is 90 degrees,
and this is 60 degrees, we know that XYZ in this case,
is going to be similar to ABC. So we would know from this
because corresponding angles are congruent, we would
know that triangle ABC is similar to triangle XYZ. And you've got to get the
order right to make sure that you have the right
corresponding angles. Y corresponds to
the 90-degree angle. X corresponds to
the 30-degree angle. A corresponds to
the 30-degree angle. So A and X are the
first two things. B and Y, which are the 90
degrees, are the second two, and then Z is the last one. So that's what we know already,
if you have three angles. But do you need three angles? If we only knew two of the
angles, would that be enough? Well, sure because if you know
two angles for a triangle, you know the third. So for example, if I
have another triangle that looks like this--
let me draw it like this-- and if I told you that only
two of the corresponding angles are congruent. So maybe this angle right here
is congruent to this angle, and that angle right there
is congruent to that angle. Is that enough to say that
these two triangles are similar? Well, sure. Because in a triangle, if
you know two of the angles, then you know what the
last angle has to be. If you know that this is 30
and you know that that is 90, then you know that this
angle has to be 60 degrees. Whatever these two angles
are, subtract them from 180, and that's going
to be this angle. So in general, in order
to show similarity, you don't have to show three
corresponding angles are congruent, you really
just have to show two. So this will be the first of
our similarity postulates. We call it angle-angle. If you could show that two
corresponding angles are congruent, then we're dealing
with similar triangles. So for example, just to
put some numbers here, if this was 30 degrees, and
we know that on this triangle, this is 90 degrees
right over here, we know that this
triangle right over here is similar to that one there. And you can really just
go to the third angle in this pretty
straightforward way. You say this third
angle is 60 degrees, so all three angles
are the same. That's one of our
constraints for similarity. Now, the other thing we
know about similarity is that the ratio
between all of the sides are going to be the same. So for example, if we have
another triangle right over here-- let me
draw another triangle-- I'll call this
triangle X, Y, and Z. And let's say that we know that
the ratio between AB and XY, we know that AB over XY-- so
the ratio between this side and this side-- notice we're not
saying that they're congruent. We're looking at
their ratio now. We're saying AB
over XY, let's say that that is equal
to BC over YZ. That is equal to BC over YZ. And that is equal to AC over XZ. So once again, this
is one of the ways that we say, hey,
this means similarity. So if you have all three
corresponding sides, the ratio between all
three corresponding sides are the same, then
we know we are dealing with similar triangles. So this is what we call
side-side-side similarity. And you don't want
to get these confused with side-side-side congruence. So these are all of our
similarity postulates or axioms or things that
we're going to assume and then we're
going to build off of them to solve problems
and prove other things. Side-side-side, when we're
talking about congruence, means that the corresponding
sides are congruent. Side-side-side for
similarity, we're saying that the ratio
between corresponding sides are going to be the same. So for example, let's say
this right over here is 10. No. Let me think of a bigger number. Let's say this is 60, this
right over here is 30, and this right over here
is 30 square roots of 3, and I just made those
numbers because we will soon learn what typical ratios
are of the sides of 30-60-90 triangles. And let's say this
one over here is 6, 3, and 3 square roots of 3. Notice AB over XY
30 square roots of 3 over 3 square roots
of 3, this will be 10. What is BC over XY? 30 divided by 3 is 10. And what is 60 divided
by 6 or AC over XZ? Well, that's going to be 10. So in general, to go from
the corresponding side here to the
corresponding side there, we always multiply
by 10 on every side. So we're not saying
they're congruent or we're not saying
the sides are the same for this
side-side-side for similarity. We're saying that we're
really just scaling them up by the same amount,
or another way to think about it, the ratio
between corresponding sides are the same. Now, what about
if we had-- let's start another triangle
right over here. Let me draw it like this. Actually, I want to leave this
here so we can have our list. So let's draw
another triangle ABC. So this is A, B,
and C. And let's say that we know that this side,
when we go to another triangle, we know that XY is AB
multiplied by some constant. So I can write it over here. XY is equal to some
constant times AB. Actually, let me make XY
bigger, so actually, it doesn't have to be. That constant could be
less than 1 in which case it would be a smaller value. But let me just do it that way. So let me just make XY
look a little bit bigger. So let's say that this
is X and that is Y. So let's say that we
know that XY over AB is equal to some constant. Or if you multiply
both sides by AB, you would get XY is some
scaled up version of AB. So maybe AB is 5, XY is 10,
then our constant would be 2. We scaled it up
by a factor of 2. And let's say we also
know that angle ABC is congruent to angle XYZ. I'll add another
point over here. So let me draw another
side right over here. So this is Z. So
let's say we also know that angle ABC
is congruent to XYZ, and let's say we know that
the ratio between BC and YZ is also this constant. The ratio between
BC and YZ is also equal to the same constant. So an example where this 5
and 10, maybe this is 3 and 6. The constant we're
kind of doubling the length of the side. So is this triangle XYZ
going to be similar? Well, if you think about it, if
XY is the same multiple of AB as YZ is a multiple of BC,
and the angle in between is congruent, there's
only one triangle we can set up over here. We're only constrained to
one triangle right over here, and so we're
completely constraining the length of this side,
and the length of this side is going to have to be that
same scale as that over there. And so we call that
side-angle-side similarity. So once again, we
saw SSS and SAS in our congruence
postulates, but we're saying something
very different here. We're saying that
in SAS, if the ratio between corresponding
sides of the true triangle are the same, so AB and XY of
one corresponding side and then another corresponding side,
so that's that second side, so that's between BC and YZ,
and the angle between them are congruent, then we're
saying it's similar. For SAS for congruency, we
said that the sides actually had to be congruent. Here we're saying that the ratio
between the corresponding sides just has to be the same. So for example SAS, just to
apply it, if I have-- let me just show some examples here. So let's say I have a
triangle here that is 3, 2, 4, and let's say we have
another triangle here that has length
9, 6, and we also know that the angle in
between are congruent so that that angle is
equal to that angle. What SAS in the
similarity world tells you is that these
triangles are definitely going to be similar
triangles, that we're actually constraining because there's
actually only one triangle we can draw a right over here. It's the triangle
where all the sides are going to have to be
scaled up by the same amount. So there's only one long
side right here that we could actually draw,
and that's going to have to be scaled
up by 3 as well. This is the only
possible triangle. If you constrain this
side you're saying, look, this is 3 times that side, this
is 3 three times that side, and the angle between
them is congruent, there's only one
triangle we could make. And we know there is a
similar triangle there where everything is scaled
up by a factor of 3, so that one triangle
we could draw has to be that one
similar triangle. So this is what we're
talking about SAS. We're not saying that this
side is congruent to that side or that side is
congruent to that side, we're saying that they're
scaled up by the same factor. If we had another triangle
that looked like this, so maybe this is 9, this is
4, and the angle between them were congruent, you
couldn't say that they're similar because this side is
scaled up by a factor of 3. This side is only scaled
up by a factor of 2. So this one right over
there you could not say that it is
necessarily similar. And likewise if you had a
triangle that had length 9 here and length 6 there,
but you did not know that these two
angles are the same, once again, you're not
constraining this enough, and you would not know that
those two triangles are necessarily similar
because you don't know that middle
angle is the same. Now, you might be
saying, well there was a few other
postulates that we had. We had AAS when we
dealt with congruency, but if you think about
it, we've already shown that two
angles by themselves are enough to show similarity. So why worry about an
angle, an angle, and a side or the ratio between a side? So why even worry about that? And we also had
angle-side-angle in congruence, but once again, we already
know the two angles are enough, so we don't need to
throw in this extra side, so we don't even need
this right over here. So these are going to be
our similarity postulates, and I want to remind
you, side-side-side, this is different than the
side-side-side for congruence. We're talking about the ratio
between corresponding sides. We're not saying that
they're actually congruent. And here, side-angle-side,
it's different than the side-angle-side
for congruence. It's this kind of
related, but here we're talking about the ratio
between the sides, not the actual measures.