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Welcome back.

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We're almost done learning all[br]the rules or laws of angles

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that we need to start[br]playing the angle game.

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So let's just teach[br]you a couple of more.

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So let's say I have two[br]parallel lines, and you may not

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know what a parallel line is[br]and I will explain

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it to you now.

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So I have one line like this --[br]you probably have an intuition

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what a parallel line means.

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That's one of my parallel[br]lines, and let me make the

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green one the other[br]parallel line.

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So parallel lines, and I'm[br]just drawing part of them.

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We assume that they keep on[br]going forever because these are

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abstract notions -- this light[br]blue line keeps going and going

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on and on and on off the screen[br]and same for this green line.

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And parallel lines are two[br]lines in the same plane.

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And a plane is just kind of[br]you can kind of use like a

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flat surface is a plane.

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We won't go into[br]three-dimensional space

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in geometry class.

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But they're on the same plane[br]and you can view this plane as

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the screen of your computer[br]right now or the piece of paper

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you're working on that never[br]intersect each other and

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they're two separate lines.

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Obviously if they were drawn[br]on top of each other then

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they intersect each[br]other everywhere.

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So it's really just two[br]lines on a plane that never

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intersect each other.

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That's a parallel line.

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If you've already learned your[br]algebra and you're familiar

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with slope, parallel lines are[br]two lines that have the

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same slope, right?

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They kind of increase or[br]decrease at the same rate.

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But they have different[br]y intercepts.

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If you don't know what[br]I'm talking about,

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don't worry about it.

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I think you know what a[br]parallel line means.

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You've seen this -- parallel[br]parking, what's parallel

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parking is when you park a car[br]right next to another car

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without having the two cars[br]intersect, because if the cars

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did intersect you would have to[br]call your insurance company.

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But anyway, so those[br]are parallel lines.

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The blue and the green[br]lines are parallel.

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And I will introduce you to[br]a new complicated geometry

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term called a transversal.

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All a transversal is is[br]another line that actually

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intersects those two lines.

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That's a transversal.

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Fancy word for something[br]very simple, transversal.

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Let me write it down just[br]to write something down.

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Transversal.

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[br]54[br]00:02:18,69 --> 00:02:23,51[br]It crosses the other two lines.

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I was thinking of pneumonics[br]for transversals, but I

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probably was thinking of[br]things inappropriate.

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[br]58[br]00:02:31,71 --> 00:02:33,81[br]Going on with the geometry.

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So we have a transversal[br]that intersects the

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two parallel lines.

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What we're going to do is think[br]of a bunch of -- and actually

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if it intersects one[br]of them it's going to

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intersect the other.

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I'll let you think about that.

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There's no way that I can draw[br]something that intersects one

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parallel line that doesn't[br]intersect the other, as long as

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this line keeps going forever.

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I think that that might be[br]pretty obvious to you.

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But what I want to do[br]is explore the angles

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of a transversal.

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So the first thing I'm[br]going to do is explore

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the corresponding angles.

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So let's say corresponding[br]angles are kind of the

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same angle at each of[br]the parallel lines.

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corresponding angles.

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They kind of play the same[br]role where the transversal

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intersects each of the lines.

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As you can imagine, and as it[br]looks from my amazingly neat

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drawing -- I'm normally not[br]this good -- that these are

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going to be equal[br]to each other.

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So if this is x, this[br]is also going to be x.

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If we know that then we could[br]use, actually the rules that we

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just learned to figure out[br]everything else about

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all of these lines.

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Because if this is x then what[br]is this going to be right here?

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What is this angle going[br]to be in magenta?

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[br]90[br]00:03:58,97 --> 00:04:00,99[br]Well, these are opposite[br]angles, right?

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They're on opposite[br]side of crossing lines

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so this is also x.

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[br]94[br]00:04:06,94 --> 00:04:08,41[br]And similar we can do[br]the same thing here.

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This is the opposite angle of[br]this angle, so this is also x.

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[br]97[br]00:04:18,58 --> 00:04:21,01[br]Let me pick a good color.

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What is yellow?

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What is this angle going to be?

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Well, just like we[br]were doing before.

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Look, we have this huge[br]angle here, right?

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This angle, this whole[br]angle is 180 degrees.

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So x and this yellow angle are[br]supplementary, so we could call

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Well, if this angle is y, then[br]this angle is opposite to y.

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So this angle is also y.

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Fascinating.

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And similarly, if we have x up[br]here and x is supplementary to

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this angle as well, right?

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So this is equal to 180 minus[br]x where it also equals y.

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And then opposite angles,[br]this is also equal to y.

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So there's all sorts of[br]geometry words and rules that

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fall out of this, and I'll[br]review them real fast but

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it's really nothing fancy.

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All I did is I started[br]off with the notion of

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corresponding angles.

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I said well, this x[br]is equal to this x.

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I said, oh well, if those are[br]equal to each other, well not

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even if -- I mean if this is x[br]and this is also x because

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they're opposite, and the[br]same thing for this.

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Then, well, if this is x and[br]this is x and those equal

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each other, as they should[br]because those are also

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corresponding angles.

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These two magenta angles[br]are playing the same role.

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They're both kind of[br]the bottom left angle.

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That's how I think about it.

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We went around, we used[br]supplementary angles to kind

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of derive well, these y[br]angles are also the same.

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This y angle is equal to[br]this y angle because

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it's corresponding.

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So corresponding angles[br]are equal to each other.

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It makes sense, they're kind[br]of playing the same role.

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The bottom right, if you look[br]at the bottom right angle.

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So corresponding[br]angles are equal.

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[br]139[br]00:06:22,87 --> 00:06:25,13[br]That's my shorthand notation.

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And we've really just[br]derived everything already.

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That's all you really[br]have to know.

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But if you wanted to kind of[br]skip a step, you also know

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the alternate interior[br]angles are equal.

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So what do I mean by[br]alternate interior angles?

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Well, the interior angles are[br]kind of the angles that are

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closer to each other in the two[br]parallel lines, but they're on

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opposite side of[br]the transversal.

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That's a very complicated way[br]of saying this orange angle and

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this magenta angle right here.

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These are alternate interior[br]angles, and we've already

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proved if this is[br]x then that is x.

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So these are alternate[br]interior angles.

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This x and then that x[br]are alternate interior.

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And actually this y and this y[br]are also alternate interior,

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and we already proved that[br]they equal each other.

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Then the last term that you'll[br]see in geometry is alternate --

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I'm not going to write the[br]whole thing -- alternate

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exterior angle.

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Alternate exterior[br]angles are also equal.

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That's the angles on the kind[br]of further away from each other

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on the parallel lines, but[br]they're still alternate.

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So an example of that is this x[br]up here and this x down here,

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right, because they're on the[br]outsides of the two parallel

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of the transversal.

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These are just fancy words,[br]but I think hopefully

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you have the intuition.

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Corresponding a angles make[br]the most sense to me.

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Then everything else proves out[br]just through opposite angles

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and supplementary angles.

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But alternate exterior is[br]that angle and that angle.

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Then the other alternate[br]exterior is this y and this y.

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Those are also equal.

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So if you know these, you know[br]pretty much everything you need

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to know about parallel lines.

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The last thing I'm going to[br]teach you in order to play the

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geometry game with full force[br]is just that the angles in a

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triangle add up to 180 degrees.

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[br]181[br]00:08:41,77 --> 00:08:45,58[br]So let me just draw a[br]triangle, a kind of

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random looking triangle.

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That's my random[br]looking triangle.

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And if this is x, this[br]is y, and this is z.

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We know that the angles of a[br]triangle -- x degrees plus y

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degrees plus z degrees are[br]equal to 180 degrees.

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So if I said that this is[br]equal to, I don't know, 30

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degrees, this is equal to,[br]I don't know, 70 degrees.

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Then what does z equal?

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Well, we would say 30 plus 70[br]plus z is equal to 180, or

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100 plus z is equal to 180.

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Subtract 100 from both sides.

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z would be equal to 80 degrees.

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We'll see variations of this[br]where you get two of the angles

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and you can use this property[br]to figure out the third.

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With everything we've now[br]learned, I think we're

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ready to kind of ease[br]into the angle game.

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I'll see you in the next video.