Welcome back.

So where we had left off, we
said, OK, we have this angle

here, can we figure out if
any of these angles

are equal to it?

Well we know that alternate
interior angles on-- this is a

transversal line right here,
and these are parallel lines.

So we know alternate interior.

So this is an interior and it's
alternate interior is here.

So we know they
equal each other.

I'm not going to draw it yet,
because sometimes if you forget

alternate interior you could
just remember, well,

corresponding angles
equal each other.

So you could say that
that angle is also

equal to this angle.

And then you can use opposite
angles again to kind of get

back to the alternate interior.

I'll show you.

The great thing about math is
it's good for people who have

trouble memorizing things,
because you have to just

memorize a few things and
then everything else just

kind of falls out of it.

But anyway.

So we figured out that
this angle is the

same as this angle.

Right?

Because they're alternate
interior angles.

And this is its
corresponding side.

And then finally, what
about this angle here?

I'm going to draw
a triple angle.

One, two, three.

What is that one equal
to on this triangle?

Well, same reason.

Alternate interior angles of
two parallel lines-- and

remember, the only reason we
can kind of make this claim is

because I told you at the
beginning that this line right

here and this line right
here are parallel.

Right?

Otherwise, you couldn't
make this claim.

But since they are alternate
interior we know that

this is the same angle.

All right.

So we now have shown that
these are similar triangles.

And I didn't have to
do all three angles.

I could have just done two, and
that should have been good

enough for you to know
that they're similar.

Because if two are the
same then the third also

has to be the same.

And now let's see if we can
use this information to

figure out our ratios.

So let's see.

Let's color the sides the
same side as the angle so

we don't get confused.

So this side is
the orange side.

Right?

This side is the blue.

This side is the red.

OK.

So we have everything
color coded.

And it might be confusing you
but it's useful, because, as

we'll see, these triangles are
actually kind of flipped.

So let's see what we can do.

So we need to figure out
this orange side here.

So this orange side
here, let's call it x.

So x equals question mark.

This orange side here
corresponds to this side here.

Right?

Because it's opposite
this angle, which is

equal to this angle.

So they're opposite
to the same angle.

So that's how we know they
correspond to each other.

So we could say x
over 6 is equal to.

And now, what other
sides do we know?

Well we know this side
here-- we know this 4 side.

Let me do it in that color.

We know this side is 4.

And since we've put x in the
numerator on the left-hand

side, and 4 is in the same
triangle as this x we're trying

to figure out, we'll put 4 in
the numerator on the

right-hand side.

4 over what?

Well what side
corresponds to 4?

What is opposite this
angle right here?

Well it's this angle.

Right?

So the corresponding side of
this side is this side-- is 5.

And now we can solve.

x is equal-- we just
multiply both sides by 6.

So you get 24 over 5.

x is equal to 24 over 5.

Not too bad.

And then we could
even go further.

We can now figure out what
this side is right here.

This magenta side.

Let's call that,
I don't know, y.

Not too creative here.

Well y corresponds
to this angle.

So y corresponds
to this 8 side.

Right?

So we could do y over 8 is
equal to-- oh, we could

do a bunch of things.

We could say 4 over 5 or we
could do-- let's do 4 over 5,

because we could do 24 over
5 over 6 and that's

kind of confusing.

So we could also do
that [UNINTELLIGIBLE]

over 4 over 5.

Multiply both sides by 8.

And you get y is equal
to 8 times 4, is what?

32 over 5.

And the reason why I did this
example is because I want

to show you that you
can't just eyeball.

Sometimes you can, if you get
good at it, but it's not always

completely obvious which sides
correspond to which sides.

It might have been tempting to
say that, I don't know, this

side corresponds to this
side or that this side

corresponds to this side.

But you really have to pay
attention to which side kind of

matches up with which angles.

So any side that matches up
with a certain angle, that same

angle in the other triangle,
whatever side is opposite that,

that's its corresponding side.

I use a lot of words, but
hopefully you have a

bit of an intuition.

Let's do another one.

First, let's take a triangle
and prove to ourselves that the

two triangles are similar.

I like these parallel lines.

Let me do two parallel
lines again.

And then this time
around-- let's see.

I'm going to draw.

There's a line.

There we go.

First, I said these
are parallel lines.

So let me mark them as such.

Parallel lines.

So what we want to do is we
want to prove that this

triangle right here is similar
to the bigger triangle-- is

similar to this triangle.

This is pretty interesting.

They actually overlap.

Right?

So first of all, do we know any
angles of the two triangles

that equal each other?

Well, sure.

They have this angle.

They actually both share the
same exact angle in common.

Right?

Because the two triangles
overlap at that point.

So what else can we figure out?

So let's see.

I mean, I don't to be
tacky without any

colors, but let's see.

We have this angle here.

And what other angles are
equal to this angle?

Well, we can use our parallel
lines and transversal of

angle rules, or theorems or
whatever, and figure it out.

Well this angle
corresponds to what?

Well, it corresponds
to this angle.

So it's equivalent.

And you got that from
your parallel lines.

Right?

So these two are the same.

And then, finally, if I have--
let me pick a good color-- if I

have this angle, draw
a triple angle here.

Same thing.

This corresponding angle is
going to be right here.

So there.

We know all of the three angles
of this triangle are the same.

So this is a similar triangle.

Let's say we know that this
side right here-- I'll give

you a little trick question.

From here to here is 5.

And from here to here is 7.

From here to here is-- I
don't know; make up a

good number-- is 12.

And from here to here
is, let me say, 6.

And I wanted to figure
out what this is.

How do we do that?

And I've further made it more
confusing by adding all

these squiggly lines.

Well, we already know
that these are two

similar triangles.

So we can use that information
to do our ratios.

So if we call this
is equal to x.

Right?

So what do we know?

We know that this whole side
corresponds to what side

on the smaller triangle?

Well, it corresponds
to this side.

Right?

It corresponds to here.

So let me draw it in
the correct color.

So if we do the orange, this
orange corresponds to this.

Right?

Well this orange corresponds
to the whole thing.

It corresponds to
this whole line.

So if we take the big
triangle, the big triangle

side is not just x.

Right?

Because that's not the whole
side of the triangle.

It's x plus 5.

That's this whole side.

Right?

x plus 5 over the corresponding
side on the smaller triangle.

Well, on the corresponding
side of the smaller

triangle it's just this.

It's over 5.

Right?

Is equal to-- and then
we could say, well, 12.

Is equal to 12, because this
corresponds to this angle

on the big triangle.

Is equal to 12 over what?

Over 6, because this is
the smaller triangle.

And then we could
solve for that.

This becomes 2.

Right?

You get x plus 5
is equal to 10.

x is equal to 5.

There you go.

That's all the time
I have for now.

I hope I helped you
understand similar triangles

just a little bit.

I'll see you soon.