0:00:00.000,0:00:00.690


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Welcome back.

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

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here, can we figure out if[br]any of these angles

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are equal to it?

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

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transversal line right here,[br]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[br]alternate interior is here.

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So we know they[br]equal each other.

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

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alternate interior you could[br]just remember, well,

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corresponding angles[br]equal each other.

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So you could say that[br]that angle is also

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equal to this angle.

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And then you can use opposite[br]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[br]it's good for people who have

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trouble memorizing things,[br]because you have to just

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memorize a few things and[br]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[br]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[br]interior angles.

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And this is its[br]corresponding side.

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And then finally, what[br]about this angle here?

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I'm going to draw[br]a triple angle.

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One, two, three.

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What is that one equal[br]to on this triangle?

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Well, same reason.

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Alternate interior angles of[br]two parallel lines-- and

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

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because I told you at the[br]beginning that this line right

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here and this line right[br]here are parallel.

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Right?

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Otherwise, you couldn't[br]make this claim.

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But since they are alternate[br]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[br]these are similar triangles.

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

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

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enough for you to know[br]that they're similar.

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Because if two are the[br]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[br]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[br]same side as the angle so

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we don't get confused.

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So this side is[br]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[br]color coded.

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

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we'll see, these triangles are[br]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[br]this orange side here.

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

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So x equals question mark.

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This orange side here[br]corresponds to this side here.

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Right?

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Because it's opposite[br]this angle, which is

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equal to this angle.

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So they're opposite[br]to the same angle.

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

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So we could say x[br]over 6 is equal to.

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And now, what other[br]sides do we know?

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Well we know this side[br]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[br]numerator on the left-hand

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

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to figure out, we'll put 4 in[br]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[br]corresponds to 4?

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What is opposite this[br]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[br]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[br]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[br]even go further.

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

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This magenta side.

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

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Not too creative here.

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Well y corresponds[br]to this angle.

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So y corresponds[br]to this 8 side.

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Right?

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So we could do y over 8 is[br]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[br]could do-- let's do 4 over 5,

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

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kind of confusing.

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So we could also do[br]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[br]to 8 times 4, is what?

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32 over 5.

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And the reason why I did this[br]example is because I want

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to show you that you[br]can't just eyeball.

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

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completely obvious which sides[br]correspond to which sides.

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

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side corresponds to this[br]side or that this side

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corresponds to this side.

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But you really have to pay[br]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[br]with a certain angle, that same

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angle in the other triangle,[br]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[br]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[br]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[br]lines again.

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And then this time[br]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[br]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[br]want to prove that this

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triangle right here is similar[br]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[br]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[br]same exact angle in common.

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Right?

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Because the two triangles[br]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[br]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[br]equal to this angle?

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Well, we can use our parallel[br]lines and transversal of

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angle rules, or theorems or[br]whatever, and figure it out.

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Well this angle[br]corresponds to what?

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Well, it corresponds[br]to this angle.

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So it's equivalent.

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And you got that from[br]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--[br]let me pick a good color-- if I

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have this angle, draw[br]a triple angle here.

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Same thing.

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This corresponding angle is[br]going to be right here.

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So there.

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We know all of the three angles[br]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[br]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[br]don't know; make up a

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good number-- is 12.

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And from here to here[br]is, let me say, 6.

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And I wanted to figure[br]out what this is.

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How do we do that?

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

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these squiggly lines.

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Well, we already know[br]that these are two

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similar triangles.

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So we can use that information[br]to do our ratios.

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So if we call this[br]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[br]corresponds to what side

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on the smaller triangle?

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Well, it corresponds[br]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[br]the correct color.

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So if we do the orange, this[br]orange corresponds to this.

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Right?

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Well this orange corresponds[br]to the whole thing.

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It corresponds to[br]this whole line.

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So if we take the big[br]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[br]side of the triangle.

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It's x plus 5.

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That's this whole side.

0:09:02.060,0:09:02.450
Right?

0:09:02.450,0:09:06.116


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x plus 5 over the corresponding[br]side on the smaller triangle.

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Well, on the corresponding[br]side of the smaller

0:09:12.660,0:09:14.630
triangle it's just this.

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It's over 5.

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Right?

0:09:17.870,0:09:22.180
Is equal to-- and then[br]we could say, well, 12.

0:09:22.180,0:09:25.740
Is equal to 12, because this[br]corresponds to this angle

0:09:25.740,0:09:27.332
on the big triangle.

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Is equal to 12 over what?

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Over 6, because this is[br]the smaller triangle.

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And then we could[br]solve for that.

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This becomes 2.

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Right?

0:09:36.860,0:09:40.936
You get x plus 5[br]is equal to 10.

0:09:40.936,0:09:43.530
x is equal to 5.

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There you go.

0:09:46.300,0:09:48.560
That's all the time[br]I have for now.

0:09:48.560,0:09:51.540
I hope I helped you[br]understand similar triangles

0:09:51.540,0:09:52.580
just a little bit.

0:09:52.580,0:09:54.720
I'll see you soon.