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Hello.

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

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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

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they're not the same thing or
they're not identical, right?

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That's the same thing
for triangles.

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So similar triangles are
two triangles that have

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all the same angles.

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For example, let me draw
two similar triangles.

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I'll try to make them look kind
of the same because they're

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supposed to look kind of the
same, but just maybe

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be different sizes.

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So that's one, and I'll draw
another one that's right here.

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I'm going to draw it a little
smaller to show you that

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they're not necessarily the
same size, they just are

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same shape essentially.

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One way I like to think about
similar triangles are they're

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just triangles that could be
kind of scaled up or down in

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size or flipped around or
rotated, but they all have

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the same angles so they're
essentially the same shape.

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For example, these two
triangles, if I were tell you

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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

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and I told you that this angle
here is equal to this angle.

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Well, a couple of things.

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You already know that this
angle's going to be equal to

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this angle, and why is that?

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Well because if two angles
are the same, then the third

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has to be the same, right?

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Because all three
angles add up to 180.

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For example, if this is x,
this is y, this one has to be

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180 minus x minus y, right?

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That's probably too
small for you to see.

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But that's the same thing here.

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If this is x and this is
y, then this angle right

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here is going to be 180
minus x minus y, right?

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So if we know that two angles
are the same in two triangles,

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so we know that the third one's
also going to be to same.

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So we could also say this angle
is identical to this angle.

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And if all the angles are the
same, then we know that we are

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dealing with similar triangles.

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What useful thing can we
now do once we know that

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a triangle is similar?

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Well, we can use that
information to kind of figure

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out some of the sides.

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So, even though they don't have
the same sides, the ratio

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of corresponding side
lengths is the same.

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I know I've just confused you.

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Let me give you an example.

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For example, let's say that
this side is -- this side is 5.

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Let's say that this side is,
I don't know, I'm just going

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to make up some number, 6.

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And let's say that this
side is 7, right?

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And let's say we know that, I
don't know, let's say we know

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that this side here is 2.

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So we know the ratio
of corresponding

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sides is the same.

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So, if we look at these two
triangles, they have completely

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different sizes but they
have corresponding sides.

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For example, this side
corresponds to this side.

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How do we know that?

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Well, in this case, they
just happen to have

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the same orientation.

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But we know that because
these sides are opposite

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the same angle, right?

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This is opposite angle y,
and then this side is

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opposite angle y again.

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This whole triangle might be
too small for you to see, but

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hopefully you're getting
what I'm saying.

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So these are
corresponding sides.

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Similarly, this side, this
blue side, and this blue side

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are corresponding sides.

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Why?

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Not because they're kind of on
the top left because we could

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have rotated this and flipped
it and whatever else.

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It's because it's
opposite the same angle.

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That's the way I always
think about triangles.

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It's a good way to think about
it, especially when you

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start doing trigonometry.

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So what does that us?

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Well, the ratio between
corresponding sides

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is always the same.

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So let's say we want to figure
out how long this side of

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the small triangle is.

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Well there's a bunch of
ways we could do it.

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We could say that the ratio of
this side to this side, so x to

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7 is going to be equal to the
ratio of this side to this side

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-- is equal to the
ratio of 2 to 5.

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And then we could solve it.

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And the only reason why we can
do this -- you can't do this

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with just random triangles, you
can only do this with

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similar triangles.

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So we could then solve for x,
multiply both sides but 7 and

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you get x is equal
to 14 over 5.

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So it's a little
bit less than 3.

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So 14 over 5, so 2.8 or
something like that,

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that equals x.

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And we could do the same thing
to figure out this yellow side.

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So if you know two triangles
are similar, you know all the

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sides of one of the triangles,
you know one of the sides of

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the other triangle, you can
figure out all the sides.

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I think I just confused
you with that comment.

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So, this side, so
let's call this y.

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you're doing one triangle's
going to be the denominator

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here, then that same
triangle has to be the

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denominator on the--.

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If one triangle is the
numerator on the left hand side

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of the equal sign, right, so
the smaller one's

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the numerator.

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Then it's also going to be the
numerator on the right hand

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side of the equal sign.

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I just want to make sure
you're consistent that way.

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If you flip it then you're
going to mess everything up.

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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

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just to do some problems.

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So let's use some of the
geometry we've already learned.

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I have two parallel lines, then
I have a line like that, then

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I have a line like this.

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What did I say, I said that the
lines are parallel, so this

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line is parallel to this line.

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And I want to know if this side
is length 5, what is -- well,

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let's say this length is length
5, let's say that this length

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is -- let me draw
another color.

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This length is, I
don't know, 8.

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I want to know what
this side is.

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Actually no, let me give you
one more side just to make sure

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you know all of one triangle.

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Let's say that this side is 6,
and what I want to do is I want

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to figure out what this side is
right here, this purple side.

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So how do we do this?

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So before we start using any of
that ratio stuff, we have to

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prove to ourselves and prove in
general, that these are

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similar triangles.

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So how can we do that?

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Let's see if we can figure
out which angles are

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equal to other angles.

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So we have this angle here.

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Is this angle equal to any
of these three angles

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in this triangle?

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Well, yeah sure.

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It's opposite this angle right
here, so this is going to be

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equal to this angle
right here, right?

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So we know that its opposite
side is it's corresponding

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side, so we know that it
corresponds to -- we don't know

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its length, but we know it
corresponds to this

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8 length, right?

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I forgot to give you
some information.

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I forgot to tell you that
this side is -- let me

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give it a neutral color.

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Let's say that this side is 4.

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Let's go back to the problem.

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So we just figured out these
two angles are the same, and

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that this is that angle's
corresponding side.

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Can we figure out any other
angles are the same?

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Let's say we know
what this angle is.

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I'm going to do kind of a
double angle measure here.

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So what angle in this triangle
-- does any angle here

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equal that angle?

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Sure.

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We know that these are parallel
lines, so we can use alternate

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interior angles to figure out
which of these angles

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equals that one.

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But I just saw the time
and I realize I'm

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running out of time.

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So I will continue this
in the next video.

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