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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.
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Right?
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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?
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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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just a little bit.
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I'll see you soon.
