Hello.
I will now introduce you to the concept of similar triangles.
Let me write that down.
So in everyday life what does similar mean?
Well, if two things are similar they're kind of the same but
they're not the same thing or they're not identical, right?
That's the same thing for triangles.
So similar triangles are two triangles that have
all the same angles.
For example, let me draw two similar triangles.
I'll try to make them look kind of the same because they're
supposed to look kind of the same, but just maybe
be different sizes.
So that's one, and I'll draw another one that's right here.
I'm going to draw it a little smaller to show you that
they're not necessarily the same size, they just are
same shape essentially.
One way I like to think about similar triangles are they're
just triangles that could be kind of scaled up or down in
size or flipped around or rotated, but they all have
the same angles so they're essentially the same shape.
For example, these two triangles, if I were tell you
that this angle -- and this is how they do it in class.
If I were to tell you this angle is equal to this angle
and I told you that this angle here is equal to this angle.
Well, a couple of things.
You already know that this angle's going to be equal to
this angle, and why is that?
Well because if two angles are the same, then the third
has to be the same, right?
Because all three angles add up to 180.
For example, if this is x, this is y, this one has to be
180 minus x minus y, right?
That's probably too small for you to see.
But that's the same thing here.
If this is x and this is y, then this angle right
here is going to be 180 minus x minus y, right?
So if we know that two angles are the same in two triangles,
so we know that the third one's also going to be to same.
So we could also say this angle is identical to this angle.
And if all the angles are the same, then we know that we are
dealing with similar triangles.
What useful thing can we now do once we know that
a triangle is similar?
Well, we can use that information to kind of figure
out some of the sides.
So, even though they don't have the same sides, the ratio
of corresponding side lengths is the same.
I know I've just confused you.
Let me give you an example.
For example, let's say that this side is -- this side is 5.
Let's say that this side is, I don't know, I'm just going
to make up some number, 6.
And let's say that this side is 7, right?
And let's say we know that, I don't know, let's say we know
that this side here is 2.
So we know the ratio of corresponding
sides is the same.
So, if we look at these two triangles, they have completely
different sizes but they have corresponding sides.
For example, this side corresponds to this side.
How do we know that?
Well, in this case, they just happen to have
the same orientation.
But we know that because these sides are opposite
the same angle, right?
This is opposite angle y, and then this side is
opposite angle y again.
This whole triangle might be too small for you to see, but
hopefully you're getting what I'm saying.
So these are corresponding sides.
Similarly, this side, this blue side, and this blue side
are corresponding sides.
Why?
Not because they're kind of on the top left because we could
have rotated this and flipped it and whatever else.
It's because it's opposite the same angle.
That's the way I always think about triangles.
It's a good way to think about it, especially when you
start doing trigonometry.
So what does that us?
Well, the ratio between corresponding sides
is always the same.
So let's say we want to figure out how long this side of
the small triangle is.
Well there's a bunch of ways we could do it.
We could say that the ratio of this side to this side, so x to
7 is going to be equal to the ratio of this side to this side
-- is equal to the ratio of 2 to 5.
And then we could solve it.
And the only reason why we can do this -- you can't do this
with just random triangles, you can only do this with
similar triangles.
So we could then solve for x, multiply both sides but 7 and
you get x is equal to 14 over 5.
So it's a little bit less than 3.
So 14 over 5, so 2.8 or something like that,
that equals x.
And we could do the same thing to figure out this yellow side.
So if you know two triangles are similar, you know all the
sides of one of the triangles, you know one of the sides of
the other triangle, you can figure out all the sides.
I think I just confused you with that comment.
So, this side, so let's call this y.
So we could do the same way.
We could say y over 6 is equal to 2 over 5, Right Make sure if
you're doing one triangle's going to be the denominator
here, then that same triangle has to be the
denominator on the--.
If one triangle is the numerator on the left hand side
of the equal sign, right, so the smaller one's
the numerator.
Then it's also going to be the numerator on the right hand
side of the equal sign.
I just want to make sure you're consistent that way.
If you flip it then you're going to mess everything up.
And then we can just solve for, so y is equal to 12 over 5.
So, let's use this information about similar triangles
just to do some problems.
So let's use some of the geometry we've already learned.
I have two parallel lines, then I have a line like that, then
I have a line like this.
What did I say, I said that the lines are parallel, so this
line is parallel to this line.
And I want to know if this side is length 5, what is -- well,
let's say this length is length 5, let's say that this length
is -- let me draw another color.
This length is, I don't know, 8.
I want to know what this side is.
Actually no, let me give you one more side just to make sure
you know all of one triangle.
Let's say that this side is 6, and what I want to do is I want
to figure out what this side is right here, this purple side.
So how do we do this?
So before we start using any of that ratio stuff, we have to
prove to ourselves and prove in general, that these are
similar triangles.
So how can we do that?
Let's see if we can figure out which angles are
equal to other angles.
So we have this angle here.
Is this angle equal to any of these three angles
in this triangle?
Well, yeah sure.
It's opposite this angle right here, so this is going to be
equal to this angle right here, right?
So we know that its opposite side is it's corresponding
side, so we know that it corresponds to -- we don't know
its length, but we know it corresponds to this
8 length, right?
I forgot to give you some information.
I forgot to tell you that this side is -- let me
give it a neutral color.
Let's say that this side is 4.
Let's go back to the problem.
So we just figured out these two angles are the same, and
that this is that angle's corresponding side.
Can we figure out any other angles are the same?
Let's say we know what this angle is.
I'm going to do kind of a double angle measure here.
So what angle in this triangle -- does any angle here
equal that angle?
Sure.
We know that these are parallel lines, so we can use alternate
interior angles to figure out which of these angles
equals that one.
But I just saw the time and I realize I'm
running out of time.
So I will continue this in the next video.