Similar triangles

Hello.

I will now introduce you to the
concept of similar triangles.

Let me write that down.

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So in everyday life what
does similar mean?

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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.

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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.

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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.

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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.

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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.

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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.

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.

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So, let's use this information
about similar triangles

just to do some problems.

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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.

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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.

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00:09:12,2 --> 00:09:15,1
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.