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Hello.

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

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Let me write that down.

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[br]6[br]00:00:14,15 --> 00:00:16,35[br]So in everyday life what[br]does similar mean?

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[br]8[br]00:00:26,89 --> 00:00:29,47[br]Well, if two things are similar[br]they're kind of the same but

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

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That's the same thing[br]for triangles.

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So similar triangles are[br]two triangles that have

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all the same angles.

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[br]14[br]00:00:50,46 --> 00:00:57,35[br]For example, let me draw[br]two similar triangles.

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

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supposed to look kind of the[br]same, but just maybe

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be different sizes.

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

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

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they're not necessarily the[br]same size, they just are

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same shape essentially.

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

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

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size or flipped around or[br]rotated, but they all have

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the same angles so they're[br]essentially the same shape.

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For example, these two[br]triangles, if I were tell you

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that this angle -- and this is[br]how they do it in class.

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[br]29[br]00:01:39,99 --> 00:01:44,27[br]If I were to tell you this[br]angle is equal to this angle

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

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[br]32[br]00:01:52,52 --> 00:01:54,01[br]Well, a couple of things.

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You already know that this[br]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[br]are the same, then the third

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has to be the same, right?

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Because all three[br]angles add up to 180.

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For example, if this is x,[br]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[br]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[br]y, then this angle right

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

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

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

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So we could also say this angle[br]is identical to this angle.

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

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dealing with similar triangles.

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What useful thing can we[br]now do once we know that

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a triangle is similar?

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Well, we can use that[br]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[br]the same sides, the ratio

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of corresponding side[br]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[br]this side is -- this side is 5.

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Let's say that this side is,[br]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[br]side is 7, right?

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

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that this side here is 2.

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[br]64[br]00:03:37,99 --> 00:03:40,18[br]So we know the ratio[br]of corresponding

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sides is the same.

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

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different sizes but they[br]have corresponding sides.

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For example, this side[br]corresponds to this side.

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How do we know that?

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Well, in this case, they[br]just happen to have

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the same orientation.

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But we know that because[br]these sides are opposite

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the same angle, right?

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This is opposite angle y,[br]and then this side is

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opposite angle y again.

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

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hopefully you're getting[br]what I'm saying.

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So these are[br]corresponding sides.

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Similarly, this side, this[br]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[br]the top left because we could

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have rotated this and flipped[br]it and whatever else.

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It's because it's[br]opposite the same angle.

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[br]86[br]00:04:32,81 --> 00:04:33,895[br]That's the way I always[br]think about triangles.

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It's a good way to think about[br]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[br]corresponding sides

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is always the same.

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

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the small triangle is.

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

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

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7 is going to be equal to the[br]ratio of this side to this side

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-- is equal to the[br]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[br]do this -- you can't do this

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with just random triangles, you[br]can only do this with

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similar triangles.

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

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you get x is equal[br]to 14 over 5.

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So it's a little[br]bit less than 3.

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So 14 over 5, so 2.8 or[br]something like that,

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that equals x.

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

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So if you know two triangles[br]are similar, you know all the

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sides of one of the triangles,[br]you know one of the sides of

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the other triangle, you can[br]figure out all the sides.

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I think I just confused[br]you with that comment.

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So, this side, so[br]let's call this y.

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

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here, then that same[br]triangle has to be the

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denominator on the--.

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If one triangle is the[br]numerator on the left hand side

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of the equal sign, right, so[br]the smaller one's

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the numerator.

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

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side of the equal sign.

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

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

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And then we can just solve for,[br]so y is equal to 12 over 5.

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[br]127[br]00:06:30,736 --> 00:06:33,92[br]So, let's use this information[br]about similar triangles

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just to do some problems.

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[br]130[br]00:06:44,75 --> 00:06:47,68[br]So let's use some of the[br]geometry we've already learned.

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I have two parallel lines, then[br]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[br]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[br]is length 5, what is -- well,

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let's say this length is length[br]5, let's say that this length

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is -- let me draw[br]another color.

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This length is, I[br]don't know, 8.

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[br]140[br]00:07:45,37 --> 00:07:48,33[br]I want to know what[br]this side is.

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Actually no, let me give you[br]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,[br]and what I want to do is I want

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to figure out what this side is[br]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[br]that ratio stuff, we have to

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prove to ourselves and prove in[br]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[br]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[br]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[br]here, so this is going to be

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equal to this angle[br]right here, right?

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

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

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its length, but we know it[br]corresponds to this

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8 length, right?

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I forgot to give you[br]some information.

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I forgot to tell you that[br]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[br]two angles are the same, and

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that this is that angle's[br]corresponding side.

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Can we figure out any other[br]angles are the same?

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Let's say we know[br]what this angle is.

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[br]172[br]00:09:12,2 --> 00:09:15,1[br]I'm going to do kind of a[br]double angle measure here.

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So what angle in this triangle[br]-- 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[br]lines, so we can use alternate

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interior angles to figure out[br]which of these angles

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equals that one.

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But I just saw the time[br]and I realize I'm

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running out of time.

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So I will continue this[br]in the next video.

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[br]