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Welcome back.

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

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here, can we figure out if
any of these angles

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are equal to it?

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

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transversal line right here,
and these are parallel lines.

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So we know alternate interior.

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

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So we know they
equal each other.

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

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alternate interior you could
just remember, well,

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corresponding angles
equal each other.

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So you could say that
that angle is also

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equal to this angle.

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

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back to the alternate interior.

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I'll show you.

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

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trouble memorizing things,
because you have to just

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memorize a few things and
then everything else just

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kind of falls out of it.

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But anyway.

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So we figured out that
this angle is the

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same as this angle.

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Right?

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Because they're alternate
interior angles.

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And this is its
corresponding side.

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And then finally, what
about this angle here?

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I'm going to draw
a triple angle.

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One, two, three.

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What is that one equal
to on this triangle?

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Well, same reason.

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Alternate interior angles of
two parallel lines-- and

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

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because I told you at the
beginning that this line right

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here and this line right
here are parallel.

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Right?

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Otherwise, you couldn't
make this claim.

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But since they are alternate
interior we know that

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this is the same angle.

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All right.

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So we now have shown that
these are similar triangles.

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

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

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enough for you to know
that they're similar.

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Because if two are the
same then the third also

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has to be the same.

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

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figure out our ratios.

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So let's see.

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

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we don't get confused.

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So this side is
the orange side.

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Right?

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This side is the blue.

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This side is the red.

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OK.

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So we have everything
color coded.

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

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

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So let's see what we can do.

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So we need to figure out
this orange side here.

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

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So x equals question mark.

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This orange side here
corresponds to this side here.

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Right?

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Because it's opposite
this angle, which is

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equal to this angle.

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So they're opposite
to the same angle.

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

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So we could say x
over 6 is equal to.

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And now, what other
sides do we know?

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

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Let me do it in that color.

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We know this side is 4.

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

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

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

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right-hand side.

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4 over what?

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Well what side
corresponds to 4?

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What is opposite this
angle right here?

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Well it's this angle.

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Right?

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

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And now we can solve.

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

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So you get 24 over 5.

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x is equal to 24 over 5.

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Not too bad.

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And then we could
even go further.

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

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This magenta side.

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

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Not too creative here.

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Well y corresponds
to this angle.

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So y corresponds
to this 8 side.

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Right?

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

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do a bunch of things.

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

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

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kind of confusing.

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So we could also do
that [UNINTELLIGIBLE]

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over 4 over 5.

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Multiply both sides by 8.

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

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32 over 5.

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

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to show you that you
can't just eyeball.

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

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completely obvious which sides
correspond to which sides.

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

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side corresponds to this
side or that this side

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corresponds to this side.

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

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matches up with which angles.

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

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

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that's its corresponding side.

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

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bit of an intuition.

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Let's do another one.

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

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two triangles are similar.

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I like these parallel lines.

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Let me do two parallel
lines again.

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

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I'm going to draw.

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There's a line.

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There we go.

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First, I said these
are parallel lines.

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So let me mark them as such.

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Parallel lines.

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

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

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similar to this triangle.

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This is pretty interesting.

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They actually overlap.

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Right?

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

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that equal each other?

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Well, sure.

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They have this angle.

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They actually both share the
same exact angle in common.

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Right?

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Because the two triangles
overlap at that point.

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So what else can we figure out?

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So let's see.

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

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colors, but let's see.

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We have this angle here.

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And what other angles are
equal to this angle?

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Well, we can use our parallel
lines and transversal of

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

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Well this angle
corresponds to what?

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Well, it corresponds
to this angle.

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So it's equivalent.

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And you got that from
your parallel lines.

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Right?

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So these two are the same.

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

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have this angle, draw
a triple angle here.

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Same thing.

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This corresponding angle is
going to be right here.

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So there.

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

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So this is a similar triangle.

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

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you a little trick question.

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From here to here is 5.

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And from here to here is 7.

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

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good number-- is 12.

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

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And I wanted to figure
out what this is.

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How do we do that?

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

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these squiggly lines.

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Well, we already know
that these are two

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similar triangles.

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So we can use that information
to do our ratios.

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So if we call this
is equal to x.

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Right?

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So what do we know?

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We know that this whole side
corresponds to what side

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on the smaller triangle?

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Well, it corresponds
to this side.

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Right?

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It corresponds to here.

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So let me draw it in
the correct color.

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

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Right?

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Well this orange corresponds
to the whole thing.

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It corresponds to
this whole line.

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So if we take the big
triangle, the big triangle

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side is not just x.

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Right?

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

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It's x plus 5.

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That's this whole side.

213
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Right?

214
00:09:02,450 --> 00:09:06,116


215
00:09:06,116 --> 00:09:11,340
x plus 5 over the corresponding
side on the smaller triangle.

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Well, on the corresponding
side of the smaller

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triangle it's just this.

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It's over 5.

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Right?

220
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Is equal to-- and then
we could say, well, 12.

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Is equal to 12, because this
corresponds to this angle

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on the big triangle.

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Is equal to 12 over what?

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Over 6, because this is
the smaller triangle.

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And then we could
solve for that.

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This becomes 2.

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Right?

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You get x plus 5
is equal to 10.

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x is equal to 5.

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There you go.

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That's all the time
I have for now.

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I hope I helped you
understand similar triangles

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00:09:51,540 --> 00:09:52,580
just a little bit.

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I'll see you soon.