[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:01.02,0:00:01.99,Default,,0000,0000,0000,,Welcome back. Dialogue: 0,0:00:01.99,0:00:06.13,Default,,0000,0000,0000,,We're almost done learning all\Nthe rules or laws of angles Dialogue: 0,0:00:06.13,0:00:09.42,Default,,0000,0000,0000,,that we need to start\Nplaying the angle game. Dialogue: 0,0:00:09.42,0:00:11.55,Default,,0000,0000,0000,,So let's just teach\Nyou a couple of more. Dialogue: 0,0:00:11.55,0:00:15.20,Default,,0000,0000,0000,,So let's say I have two\Nparallel lines, and you may not Dialogue: 0,0:00:15.20,0:00:17.70,Default,,0000,0000,0000,,know what a parallel line is\Nand I will explain Dialogue: 0,0:00:17.70,0:00:18.85,Default,,0000,0000,0000,,it to you now. Dialogue: 0,0:00:18.85,0:00:23.57,Default,,0000,0000,0000,,So I have one line like this --\Nyou probably have an intuition Dialogue: 0,0:00:23.57,0:00:26.33,Default,,0000,0000,0000,,what a parallel line means. Dialogue: 0,0:00:26.33,0:00:29.14,Default,,0000,0000,0000,,That's one of my parallel\Nlines, and let me make the Dialogue: 0,0:00:29.14,0:00:32.54,Default,,0000,0000,0000,,green one the other\Nparallel line. Dialogue: 0,0:00:32.54,0:00:34.91,Default,,0000,0000,0000,,So parallel lines, and I'm\Njust drawing part of them. Dialogue: 0,0:00:34.91,0:00:37.32,Default,,0000,0000,0000,,We assume that they keep on\Ngoing forever because these are Dialogue: 0,0:00:37.32,0:00:42.08,Default,,0000,0000,0000,,abstract notions -- this light\Nblue line keeps going and going Dialogue: 0,0:00:42.08,0:00:44.88,Default,,0000,0000,0000,,on and on and on off the screen\Nand same for this green line. Dialogue: 0,0:00:44.88,0:00:47.93,Default,,0000,0000,0000,,And parallel lines are two\Nlines in the same plane. Dialogue: 0,0:00:47.93,0:00:50.31,Default,,0000,0000,0000,,And a plane is just kind of\Nyou can kind of use like a Dialogue: 0,0:00:50.31,0:00:53.27,Default,,0000,0000,0000,,flat surface is a plane. Dialogue: 0,0:00:53.27,0:00:56.63,Default,,0000,0000,0000,,We won't go into\Nthree-dimensional space Dialogue: 0,0:00:56.63,0:00:58.45,Default,,0000,0000,0000,,in geometry class. Dialogue: 0,0:00:58.45,0:01:00.99,Default,,0000,0000,0000,,But they're on the same plane\Nand you can view this plane as Dialogue: 0,0:01:00.99,0:01:03.13,Default,,0000,0000,0000,,the screen of your computer\Nright now or the piece of paper Dialogue: 0,0:01:03.13,0:01:05.61,Default,,0000,0000,0000,,you're working on that never\Nintersect each other and Dialogue: 0,0:01:05.61,0:01:06.96,Default,,0000,0000,0000,,they're two separate lines. Dialogue: 0,0:01:06.96,0:01:09.62,Default,,0000,0000,0000,,Obviously if they were drawn\Non top of each other then Dialogue: 0,0:01:09.62,0:01:11.41,Default,,0000,0000,0000,,they intersect each\Nother everywhere. Dialogue: 0,0:01:11.41,0:01:13.50,Default,,0000,0000,0000,,So it's really just two\Nlines on a plane that never Dialogue: 0,0:01:13.50,0:01:14.64,Default,,0000,0000,0000,,intersect each other. Dialogue: 0,0:01:14.64,0:01:15.84,Default,,0000,0000,0000,,That's a parallel line. Dialogue: 0,0:01:15.84,0:01:18.21,Default,,0000,0000,0000,,If you've already learned your\Nalgebra and you're familiar Dialogue: 0,0:01:18.21,0:01:21.19,Default,,0000,0000,0000,,with slope, parallel lines are\Ntwo lines that have the Dialogue: 0,0:01:21.19,0:01:22.43,Default,,0000,0000,0000,,same slope, right? Dialogue: 0,0:01:22.43,0:01:26.16,Default,,0000,0000,0000,,They kind of increase or\Ndecrease at the same rate. Dialogue: 0,0:01:26.16,0:01:27.54,Default,,0000,0000,0000,,But they have different\Ny intercepts. Dialogue: 0,0:01:27.54,0:01:28.80,Default,,0000,0000,0000,,If you don't know what\NI'm talking about, Dialogue: 0,0:01:28.80,0:01:29.51,Default,,0000,0000,0000,,don't worry about it. Dialogue: 0,0:01:29.51,0:01:31.67,Default,,0000,0000,0000,,I think you know what a\Nparallel line means. Dialogue: 0,0:01:31.67,0:01:33.84,Default,,0000,0000,0000,,You've seen this -- parallel\Nparking, what's parallel Dialogue: 0,0:01:33.84,0:01:37.08,Default,,0000,0000,0000,,parking is when you park a car\Nright next to another car Dialogue: 0,0:01:37.08,0:01:39.97,Default,,0000,0000,0000,,without having the two cars\Nintersect, because if the cars Dialogue: 0,0:01:39.97,0:01:42.69,Default,,0000,0000,0000,,did intersect you would have to\Ncall your insurance company. Dialogue: 0,0:01:42.69,0:01:44.71,Default,,0000,0000,0000,,But anyway, so those\Nare parallel lines. Dialogue: 0,0:01:44.71,0:01:48.44,Default,,0000,0000,0000,,The blue and the green\Nlines are parallel. Dialogue: 0,0:01:48.44,0:01:51.21,Default,,0000,0000,0000,,And I will introduce you to\Na new complicated geometry Dialogue: 0,0:01:51.21,0:01:54.05,Default,,0000,0000,0000,,term called a transversal. Dialogue: 0,0:01:54.05,0:01:58.80,Default,,0000,0000,0000,,All a transversal is is\Nanother line that actually Dialogue: 0,0:01:58.80,0:02:01.94,Default,,0000,0000,0000,,intersects those two lines. Dialogue: 0,0:02:01.94,0:02:03.32,Default,,0000,0000,0000,,That's a transversal. Dialogue: 0,0:02:03.32,0:02:07.31,Default,,0000,0000,0000,,Fancy word for something\Nvery simple, transversal. Dialogue: 0,0:02:07.31,0:02:10.37,Default,,0000,0000,0000,,Let me write it down just\Nto write something down. Dialogue: 0,0:02:10.37,0:02:10.74,Default,,0000,0000,0000,,Transversal. Dialogue: 0,0:02:10.74,0:02:18.69,Default,,0000,0000,0000,,\N54\N00:02:18,69 --> 00:02:23,51\NIt crosses the other two lines. Dialogue: 0,0:02:23.51,0:02:25.64,Default,,0000,0000,0000,,I was thinking of pneumonics\Nfor transversals, but I Dialogue: 0,0:02:25.64,0:02:27.39,Default,,0000,0000,0000,,probably was thinking of\Nthings inappropriate. Dialogue: 0,0:02:27.39,0:02:31.71,Default,,0000,0000,0000,,\N58\N00:02:31,71 --> 00:02:33,81\NGoing on with the geometry. Dialogue: 0,0:02:33.81,0:02:36.71,Default,,0000,0000,0000,,So we have a transversal\Nthat intersects the Dialogue: 0,0:02:36.71,0:02:38.66,Default,,0000,0000,0000,,two parallel lines. Dialogue: 0,0:02:38.66,0:02:40.91,Default,,0000,0000,0000,,What we're going to do is think\Nof a bunch of -- and actually Dialogue: 0,0:02:40.91,0:02:42.06,Default,,0000,0000,0000,,if it intersects one\Nof them it's going to Dialogue: 0,0:02:42.06,0:02:43.32,Default,,0000,0000,0000,,intersect the other. Dialogue: 0,0:02:43.32,0:02:44.38,Default,,0000,0000,0000,,I'll let you think about that. Dialogue: 0,0:02:44.38,0:02:46.94,Default,,0000,0000,0000,,There's no way that I can draw\Nsomething that intersects one Dialogue: 0,0:02:46.94,0:02:49.75,Default,,0000,0000,0000,,parallel line that doesn't\Nintersect the other, as long as Dialogue: 0,0:02:49.75,0:02:51.80,Default,,0000,0000,0000,,this line keeps going forever. Dialogue: 0,0:02:51.80,0:02:53.79,Default,,0000,0000,0000,,I think that that might be\Npretty obvious to you. Dialogue: 0,0:02:53.79,0:02:56.69,Default,,0000,0000,0000,,But what I want to do\Nis explore the angles Dialogue: 0,0:02:56.69,0:02:58.64,Default,,0000,0000,0000,,of a transversal. Dialogue: 0,0:02:58.64,0:03:03.18,Default,,0000,0000,0000,,So the first thing I'm\Ngoing to do is explore Dialogue: 0,0:03:03.18,0:03:05.49,Default,,0000,0000,0000,,the corresponding angles. Dialogue: 0,0:03:05.49,0:03:08.50,Default,,0000,0000,0000,,So let's say corresponding\Nangles are kind of the Dialogue: 0,0:03:08.50,0:03:10.89,Default,,0000,0000,0000,,same angle at each of\Nthe parallel lines. Dialogue: 0,0:03:17.24,0:03:20.26,Default,,0000,0000,0000,,corresponding angles. Dialogue: 0,0:03:20.26,0:03:22.89,Default,,0000,0000,0000,,They kind of play the same\Nrole where the transversal Dialogue: 0,0:03:22.89,0:03:24.83,Default,,0000,0000,0000,,intersects each of the lines. Dialogue: 0,0:03:24.83,0:03:28.82,Default,,0000,0000,0000,,As you can imagine, and as it\Nlooks from my amazingly neat Dialogue: 0,0:03:28.82,0:03:31.39,Default,,0000,0000,0000,,drawing -- I'm normally not\Nthis good -- that these are Dialogue: 0,0:03:31.39,0:03:32.78,Default,,0000,0000,0000,,going to be equal\Nto each other. Dialogue: 0,0:03:32.78,0:03:38.50,Default,,0000,0000,0000,,So if this is x, this\Nis also going to be x. Dialogue: 0,0:03:38.50,0:03:42.50,Default,,0000,0000,0000,,If we know that then we could\Nuse, actually the rules that we Dialogue: 0,0:03:42.50,0:03:44.51,Default,,0000,0000,0000,,just learned to figure out\Neverything else about Dialogue: 0,0:03:44.51,0:03:46.39,Default,,0000,0000,0000,,all of these lines. Dialogue: 0,0:03:46.39,0:03:51.74,Default,,0000,0000,0000,,Because if this is x then what\Nis this going to be right here? Dialogue: 0,0:03:51.74,0:03:55.26,Default,,0000,0000,0000,,What is this angle going\Nto be in magenta? Dialogue: 0,0:03:55.26,0:03:58.97,Default,,0000,0000,0000,,\N90\N00:03:58,97 --> 00:04:00,99\NWell, these are opposite\Nangles, right? Dialogue: 0,0:04:00.99,0:04:02.78,Default,,0000,0000,0000,,They're on opposite\Nside of crossing lines Dialogue: 0,0:04:02.78,0:04:03.81,Default,,0000,0000,0000,,so this is also x. Dialogue: 0,0:04:03.81,0:04:06.94,Default,,0000,0000,0000,,\N94\N00:04:06,94 --> 00:04:08,41\NAnd similar we can do\Nthe same thing here. Dialogue: 0,0:04:08.41,0:04:12.03,Default,,0000,0000,0000,,This is the opposite angle of\Nthis angle, so this is also x. Dialogue: 0,0:04:12.03,0:04:18.58,Default,,0000,0000,0000,,\N97\N00:04:18,58 --> 00:04:21,01\NLet me pick a good color. Dialogue: 0,0:04:21.01,0:04:23.52,Default,,0000,0000,0000,,What is yellow? Dialogue: 0,0:04:23.52,0:04:26.18,Default,,0000,0000,0000,,What is this angle going to be? Dialogue: 0,0:04:26.18,0:04:27.31,Default,,0000,0000,0000,,Well, just like we\Nwere doing before. Dialogue: 0,0:04:27.31,0:04:30.09,Default,,0000,0000,0000,,Look, we have this huge\Nangle here, right? Dialogue: 0,0:04:30.09,0:04:33.91,Default,,0000,0000,0000,,This angle, this whole\Nangle is 180 degrees. Dialogue: 0,0:04:33.91,0:04:38.86,Default,,0000,0000,0000,,So x and this yellow angle are\Nsupplementary, so we could call Dialogue: 0,0:04:49.30,0:04:53.26,Default,,0000,0000,0000,,Well, if this angle is y, then\Nthis angle is opposite to y. Dialogue: 0,0:04:53.26,0:04:57.10,Default,,0000,0000,0000,,So this angle is also y. Dialogue: 0,0:04:57.10,0:04:58.56,Default,,0000,0000,0000,,Fascinating. Dialogue: 0,0:04:58.56,0:05:03.22,Default,,0000,0000,0000,,And similarly, if we have x up\Nhere and x is supplementary to Dialogue: 0,0:05:03.22,0:05:05.92,Default,,0000,0000,0000,,this angle as well, right? Dialogue: 0,0:05:05.92,0:05:10.60,Default,,0000,0000,0000,,So this is equal to 180 minus\Nx where it also equals y. Dialogue: 0,0:05:10.60,0:05:15.33,Default,,0000,0000,0000,,And then opposite angles,\Nthis is also equal to y. Dialogue: 0,0:05:15.33,0:05:19.17,Default,,0000,0000,0000,,So there's all sorts of\Ngeometry words and rules that Dialogue: 0,0:05:19.17,0:05:21.17,Default,,0000,0000,0000,,fall out of this, and I'll\Nreview them real fast but Dialogue: 0,0:05:21.17,0:05:22.09,Default,,0000,0000,0000,,it's really nothing fancy. Dialogue: 0,0:05:22.09,0:05:23.85,Default,,0000,0000,0000,,All I did is I started\Noff with the notion of Dialogue: 0,0:05:23.85,0:05:24.85,Default,,0000,0000,0000,,corresponding angles. Dialogue: 0,0:05:24.85,0:05:28.32,Default,,0000,0000,0000,,I said well, this x\Nis equal to this x. Dialogue: 0,0:05:28.32,0:05:32.35,Default,,0000,0000,0000,,I said, oh well, if those are\Nequal to each other, well not Dialogue: 0,0:05:32.35,0:05:34.81,Default,,0000,0000,0000,,even if -- I mean if this is x\Nand this is also x because Dialogue: 0,0:05:34.81,0:05:37.59,Default,,0000,0000,0000,,they're opposite, and the\Nsame thing for this. Dialogue: 0,0:05:37.59,0:05:40.26,Default,,0000,0000,0000,,Then, well, if this is x and\Nthis is x and those equal Dialogue: 0,0:05:40.26,0:05:42.75,Default,,0000,0000,0000,,each other, as they should\Nbecause those are also Dialogue: 0,0:05:42.75,0:05:44.75,Default,,0000,0000,0000,,corresponding angles. Dialogue: 0,0:05:44.75,0:05:48.31,Default,,0000,0000,0000,,These two magenta angles\Nare playing the same role. Dialogue: 0,0:05:48.31,0:05:50.27,Default,,0000,0000,0000,,They're both kind of\Nthe bottom left angle. Dialogue: 0,0:05:50.27,0:05:51.97,Default,,0000,0000,0000,,That's how I think about it. Dialogue: 0,0:05:51.97,0:05:54.42,Default,,0000,0000,0000,,We went around, we used\Nsupplementary angles to kind Dialogue: 0,0:05:54.42,0:05:56.82,Default,,0000,0000,0000,,of derive well, these y\Nangles are also the same. Dialogue: 0,0:06:00.29,0:06:02.27,Default,,0000,0000,0000,,This y angle is equal to\Nthis y angle because Dialogue: 0,0:06:02.27,0:06:03.66,Default,,0000,0000,0000,,it's corresponding. Dialogue: 0,0:06:03.66,0:06:06.80,Default,,0000,0000,0000,,So corresponding angles\Nare equal to each other. Dialogue: 0,0:06:06.80,0:06:09.82,Default,,0000,0000,0000,,It makes sense, they're kind\Nof playing the same role. Dialogue: 0,0:06:09.82,0:06:12.27,Default,,0000,0000,0000,,The bottom right, if you look\Nat the bottom right angle. Dialogue: 0,0:06:12.27,0:06:14.02,Default,,0000,0000,0000,,So corresponding\Nangles are equal. Dialogue: 0,0:06:14.02,0:06:22.87,Default,,0000,0000,0000,,\N139\N00:06:22,87 --> 00:06:25,13\NThat's my shorthand notation. Dialogue: 0,0:06:25.13,0:06:27.36,Default,,0000,0000,0000,,And we've really just\Nderived everything already. Dialogue: 0,0:06:27.36,0:06:28.65,Default,,0000,0000,0000,,That's all you really\Nhave to know. Dialogue: 0,0:06:28.65,0:06:31.04,Default,,0000,0000,0000,,But if you wanted to kind of\Nskip a step, you also know Dialogue: 0,0:06:31.04,0:06:46.53,Default,,0000,0000,0000,,the alternate interior\Nangles are equal. Dialogue: 0,0:06:46.53,0:06:50.32,Default,,0000,0000,0000,,So what do I mean by\Nalternate interior angles? Dialogue: 0,0:06:50.32,0:06:53.98,Default,,0000,0000,0000,,Well, the interior angles are\Nkind of the angles that are Dialogue: 0,0:06:53.98,0:06:57.56,Default,,0000,0000,0000,,closer to each other in the two\Nparallel lines, but they're on Dialogue: 0,0:06:57.56,0:06:59.41,Default,,0000,0000,0000,,opposite side of\Nthe transversal. Dialogue: 0,0:06:59.41,0:07:01.85,Default,,0000,0000,0000,,That's a very complicated way\Nof saying this orange angle and Dialogue: 0,0:07:01.85,0:07:03.30,Default,,0000,0000,0000,,this magenta angle right here. Dialogue: 0,0:07:03.30,0:07:05.76,Default,,0000,0000,0000,,These are alternate interior\Nangles, and we've already Dialogue: 0,0:07:05.76,0:07:08.63,Default,,0000,0000,0000,,proved if this is\Nx then that is x. Dialogue: 0,0:07:08.63,0:07:11.42,Default,,0000,0000,0000,,So these are alternate\Ninterior angles. Dialogue: 0,0:07:11.42,0:07:17.57,Default,,0000,0000,0000,,This x and then that x\Nare alternate interior. Dialogue: 0,0:07:17.57,0:07:22.22,Default,,0000,0000,0000,,And actually this y and this y\Nare also alternate interior, Dialogue: 0,0:07:22.22,0:07:24.12,Default,,0000,0000,0000,,and we already proved that\Nthey equal each other. Dialogue: 0,0:07:24.12,0:07:29.52,Default,,0000,0000,0000,,Then the last term that you'll\Nsee in geometry is alternate -- Dialogue: 0,0:07:29.52,0:07:31.36,Default,,0000,0000,0000,,I'm not going to write the\Nwhole thing -- alternate Dialogue: 0,0:07:31.36,0:07:33.80,Default,,0000,0000,0000,,exterior angle. Dialogue: 0,0:07:33.80,0:07:37.76,Default,,0000,0000,0000,,Alternate exterior\Nangles are also equal. Dialogue: 0,0:07:37.76,0:07:40.97,Default,,0000,0000,0000,,That's the angles on the kind\Nof further away from each other Dialogue: 0,0:07:40.97,0:07:43.27,Default,,0000,0000,0000,,on the parallel lines, but\Nthey're still alternate. Dialogue: 0,0:07:43.27,0:07:48.79,Default,,0000,0000,0000,,So an example of that is this x\Nup here and this x down here, Dialogue: 0,0:07:48.79,0:07:53.54,Default,,0000,0000,0000,,right, because they're on the\Noutsides of the two parallel Dialogue: 0,0:07:58.47,0:07:59.68,Default,,0000,0000,0000,,of the transversal. Dialogue: 0,0:07:59.68,0:08:01.72,Default,,0000,0000,0000,,These are just fancy words,\Nbut I think hopefully Dialogue: 0,0:08:01.72,0:08:03.77,Default,,0000,0000,0000,,you have the intuition. Dialogue: 0,0:08:03.77,0:08:06.41,Default,,0000,0000,0000,,Corresponding a angles make\Nthe most sense to me. Dialogue: 0,0:08:06.41,0:08:09.18,Default,,0000,0000,0000,,Then everything else proves out\Njust through opposite angles Dialogue: 0,0:08:09.18,0:08:10.45,Default,,0000,0000,0000,,and supplementary angles. Dialogue: 0,0:08:10.45,0:08:18.15,Default,,0000,0000,0000,,But alternate exterior is\Nthat angle and that angle. Dialogue: 0,0:08:18.15,0:08:22.88,Default,,0000,0000,0000,,Then the other alternate\Nexterior is this y and this y. Dialogue: 0,0:08:22.88,0:08:23.87,Default,,0000,0000,0000,,Those are also equal. Dialogue: 0,0:08:23.87,0:08:27.15,Default,,0000,0000,0000,,So if you know these, you know\Npretty much everything you need Dialogue: 0,0:08:27.15,0:08:29.19,Default,,0000,0000,0000,,to know about parallel lines. Dialogue: 0,0:08:29.19,0:08:32.30,Default,,0000,0000,0000,,The last thing I'm going to\Nteach you in order to play the Dialogue: 0,0:08:32.30,0:08:35.78,Default,,0000,0000,0000,,geometry game with full force\Nis just that the angles in a Dialogue: 0,0:08:35.78,0:08:38.14,Default,,0000,0000,0000,,triangle add up to 180 degrees. Dialogue: 0,0:08:38.14,0:08:41.77,Default,,0000,0000,0000,,\N181\N00:08:41,77 --> 00:08:45,58\NSo let me just draw a\Ntriangle, a kind of Dialogue: 0,0:08:45.58,0:08:48.58,Default,,0000,0000,0000,,random looking triangle. Dialogue: 0,0:08:48.58,0:08:51.30,Default,,0000,0000,0000,,That's my random\Nlooking triangle. Dialogue: 0,0:08:51.30,0:08:57.69,Default,,0000,0000,0000,,And if this is x, this\Nis y, and this is z. Dialogue: 0,0:08:57.69,0:09:01.38,Default,,0000,0000,0000,,We know that the angles of a\Ntriangle -- x degrees plus y Dialogue: 0,0:09:01.38,0:09:06.91,Default,,0000,0000,0000,,degrees plus z degrees are\Nequal to 180 degrees. Dialogue: 0,0:09:06.91,0:09:09.58,Default,,0000,0000,0000,,So if I said that this is\Nequal to, I don't know, 30 Dialogue: 0,0:09:09.58,0:09:15.24,Default,,0000,0000,0000,,degrees, this is equal to,\NI don't know, 70 degrees. Dialogue: 0,0:09:15.24,0:09:16.17,Default,,0000,0000,0000,,Then what does z equal? Dialogue: 0,0:09:16.17,0:09:23.65,Default,,0000,0000,0000,,Well, we would say 30 plus 70\Nplus z is equal to 180, or Dialogue: 0,0:09:23.65,0:09:27.74,Default,,0000,0000,0000,,100 plus z is equal to 180. Dialogue: 0,0:09:27.74,0:09:29.15,Default,,0000,0000,0000,,Subtract 100 from both sides. Dialogue: 0,0:09:29.15,0:09:33.48,Default,,0000,0000,0000,,z would be equal to 80 degrees. Dialogue: 0,0:09:33.48,0:09:36.15,Default,,0000,0000,0000,,We'll see variations of this\Nwhere you get two of the angles Dialogue: 0,0:09:36.15,0:09:39.25,Default,,0000,0000,0000,,and you can use this property\Nto figure out the third. Dialogue: 0,0:09:39.25,0:09:41.45,Default,,0000,0000,0000,,With everything we've now\Nlearned, I think we're Dialogue: 0,0:09:41.45,0:09:45.29,Default,,0000,0000,0000,,ready to kind of ease\Ninto the angle game. Dialogue: 0,0:09:45.29,0:09:47.51,Default,,0000,0000,0000,,I'll see you in the next video.