0:00:00.000,0:00:00.490 0:00:00.490,0:00:02.710 In this video we're going to[br]think a little bit about 0:00:02.710,0:00:11.580 parallel lines, and other lines[br]that intersect the parallel 0:00:11.580,0:00:13.780 lines, and we call[br]those transversals. 0:00:13.780,0:00:16.810 So first let's think about[br]what a parallel or what 0:00:16.810,0:00:18.490 parallel lines are. 0:00:18.490,0:00:21.700 So one definition we could use,[br]and I think that'll work well 0:00:21.700,0:00:24.220 for the purposes of this video,[br]are they're two lines that 0:00:24.220,0:00:25.660 sit in the same plane. 0:00:25.660,0:00:29.090 And when I talk about a plane,[br]I'm talking about a, you can 0:00:29.090,0:00:32.490 imagine a flat two-dimensional[br]surface like this screen -- 0:00:32.490,0:00:33.910 this screen is a plane. 0:00:33.910,0:00:37.730 So two lines that sit in a[br]plane that never intersect. 0:00:37.730,0:00:41.570 So this line -- I'll try my[br]best to draw it -- and imagine 0:00:41.570,0:00:43.750 the line just keeps going in[br]that direction and that 0:00:43.750,0:00:47.280 direction -- let me do another[br]one in a different color -- 0:00:47.280,0:00:52.050 and this line right[br]here are parallel. 0:00:52.050,0:00:53.690 They will never intersect. 0:00:53.690,0:00:55.660 If you assume that I drew it[br]straight enough and that 0:00:55.660,0:00:58.000 they're going in the exact[br]same direction, they 0:00:58.000,0:00:59.840 will never intersect. 0:00:59.840,0:01:02.070 And so if you think about what[br]types of lines are not 0:01:02.070,0:01:07.840 parallel, well, this green line[br]and this pink line 0:01:07.840,0:01:08.940 are not parallel. 0:01:08.940,0:01:11.940 They clearly intersect[br]at some point. 0:01:11.940,0:01:15.350 So these two guys are parallel[br]right over here, and sometimes 0:01:15.350,0:01:18.690 it's specified, sometimes[br]people will draw an arrow going 0:01:18.690,0:01:20.900 in the same direction to show[br]that those two lines 0:01:20.900,0:01:21.840 are parallel. 0:01:21.840,0:01:24.400 If there are multiple parallel[br]lines, they might do two arrows 0:01:24.400,0:01:25.760 and two arrows or whatever. 0:01:25.760,0:01:27.270 But you just have to say[br]OK, these lines will 0:01:27.270,0:01:28.550 never intersect. 0:01:28.550,0:01:31.060 What we want to think about[br]is what happens when 0:01:31.060,0:01:36.200 these parallel lines are[br]intersected by a third line. 0:01:36.200,0:01:37.820 Let me draw the[br]third line here. 0:01:37.820,0:01:41.690 So third line like this. 0:01:41.690,0:01:45.970 And we call that, right there,[br]the third line that intersects 0:01:45.970,0:01:52.170 the parallel lines we[br]call a transversal line. 0:01:52.170,0:01:56.150 Because it tranverses[br]the two parallel lines. 0:01:56.150,0:01:59.230 Now whenever you have a[br]transversal crossing parallel 0:01:59.230,0:02:02.190 lines, you have an interesting[br]relationship between 0:02:02.190,0:02:03.320 the angles form. 0:02:03.320,0:02:05.660 Now this shows up on a lot[br]of standardized tests. 0:02:05.660,0:02:09.200 It's kind of a core type[br]of a geometry problem. 0:02:09.200,0:02:12.450 So it's a good thing to really[br]get clear in our heads. 0:02:12.450,0:02:15.620 So the first thing to realize[br]is if these lines are parallel, 0:02:15.620,0:02:18.350 we're going to assume these[br]lines are parallel, then we 0:02:18.350,0:02:21.760 have corresponding angles[br]are going to be the same. 0:02:21.760,0:02:25.820 What I mean by corresponding[br]angles are I guess you could 0:02:25.820,0:02:28.840 think there are four angles[br]that get formed when this 0:02:28.840,0:02:31.195 purple line or this[br]magenta line intersects 0:02:31.195,0:02:32.350 this yellow line. 0:02:32.350,0:02:38.070 You have this angle up here[br]that I've specified in green, 0:02:38.070,0:02:42.970 you have -- let me do another[br]one in orange -- you have this 0:02:42.970,0:02:48.280 angle right here in orange, you[br]have this angle right here in 0:02:48.280,0:02:52.600 this other shade of green, and[br]then you have this angle 0:02:52.600,0:02:55.290 right here -- right there[br]that I've made in that 0:02:55.290,0:02:56.930 bluish-purplish color. 0:02:56.930,0:02:58.790 So those are the four angles. 0:02:58.790,0:03:01.680 So when we talk about[br]corresponding angles, we're 0:03:01.680,0:03:04.770 talking about, for example,[br]this top right angle in green 0:03:04.770,0:03:08.930 up here, that corresponds to[br]this top right angle in, what 0:03:08.930,0:03:12.040 I can draw it in that same[br]green, right over here. 0:03:12.040,0:03:14.570 These two angles[br]are corresponding. 0:03:14.570,0:03:17.990 These two are corresponding[br]angles and they're 0:03:17.990,0:03:19.520 going to be equal. 0:03:19.520,0:03:20.820 These are equal angles. 0:03:20.820,0:03:24.410 If this is -- I'll make up[br]a number -- if this is 70 0:03:24.410,0:03:27.880 degrees, then this angle[br]right here is also 0:03:27.880,0:03:29.410 going to be 70 degrees. 0:03:29.410,0:03:32.000 And if you just think about it,[br]or if you even play with 0:03:32.000,0:03:35.150 toothpicks or something, and[br]you keep changing the direction 0:03:35.150,0:03:38.140 of this transversal line,[br]you'll see that it actually 0:03:38.140,0:03:40.750 looks like they should[br]always be equal. 0:03:40.750,0:03:43.200 If I were to take -- let me[br]draw two other parallel 0:03:43.200,0:03:45.980 lines, let me show maybe[br]a more extreme example. 0:03:45.980,0:03:50.350 So if I have two other parallel[br]lines like that, and then let 0:03:50.350,0:03:57.340 me make a transversal that[br]forms a smaller -- it's even a 0:03:57.340,0:03:59.930 smaller angle here -- you see[br]that this angle right here 0:03:59.930,0:04:02.070 looks the same as that angle. 0:04:02.070,0:04:05.340 Those are corresponding angles[br]and they will be equivalent. 0:04:05.340,0:04:08.330 From this perspective it's kind[br]of the top right angle and each 0:04:08.330,0:04:10.430 intersection is the same. 0:04:10.430,0:04:13.600 Now the same is true of the[br]other corresponding angles. 0:04:13.600,0:04:16.660 This angle right here in this[br]example, it's the top left 0:04:16.660,0:04:21.120 angle will be the same as the[br]top left angle right over here. 0:04:21.120,0:04:27.080 This bottom left angle will[br]be the same down here. 0:04:27.080,0:04:30.000 If this right here is 70[br]degrees, then this down here 0:04:30.000,0:04:32.040 will also be 70 degrees. 0:04:32.040,0:04:36.040 And then finally, of course,[br]this angle and this angle 0:04:36.040,0:04:37.990 will also be the same. 0:04:37.990,0:04:41.520 So corresponding angles -- let[br]me write these -- these are 0:04:41.520,0:04:43.170 corresponding angles[br]are congruent. 0:04:43.170,0:04:46.640 0:04:46.640,0:04:55.180 Corresponding angles are equal. 0:04:55.180,0:04:57.050 And that and that are[br]corresponding, that and 0:04:57.050,0:04:59.400 that, that and that,[br]and that and that. 0:04:59.400,0:05:04.600 Now, the next set of equal[br]angles to realize are sometimes 0:05:04.600,0:05:06.610 they're called vertical angles,[br]sometimes they're called 0:05:06.610,0:05:08.440 opposite angles. 0:05:08.440,0:05:11.700 But if you take this angle[br]right here, the angle that is 0:05:11.700,0:05:15.060 vertical to it or is opposite[br]as you go right across the 0:05:15.060,0:05:18.650 point of intersection is this[br]angle right here, and that is 0:05:18.650,0:05:20.580 going to be the same thing. 0:05:20.580,0:05:23.860 So we could say opposite -- I[br]like opposite because it's not 0:05:23.860,0:05:25.720 always in the vertical[br]direction, sometimes it's in 0:05:25.720,0:05:27.650 the horizontal direction, but[br]sometimes they're referred 0:05:27.650,0:05:29.400 to as vertical angles. 0:05:29.400,0:05:37.370 Opposite or vertical[br]angles are also equal. 0:05:37.370,0:05:40.940 So if that's 70 degrees, then[br]this is also 70 degrees. 0:05:40.940,0:05:43.980 And if this is 70 degrees,[br]then this right here 0:05:43.980,0:05:46.710 is also 70 degrees. 0:05:46.710,0:05:49.240 So it's interesting, if that's[br]70 degrees and that's 70 0:05:49.240,0:05:52.230 degrees, and if this is 70[br]degrees and that is also 70 0:05:52.230,0:05:55.750 degrees, so no matter what this[br]is, this will also be the same 0:05:55.750,0:05:58.060 thing because this is[br]the same as that, that 0:05:58.060,0:05:59.770 is the same as that. 0:05:59.770,0:06:07.180 Now, the last one that you need[br]to I guess kind of realize are 0:06:07.180,0:06:09.870 the relationship between[br]this orange angle and this 0:06:09.870,0:06:11.860 green angle right there. 0:06:11.860,0:06:17.890 You can see that when you add[br]up the angles, you go halfway 0:06:17.890,0:06:19.710 around a circle, right? 0:06:19.710,0:06:22.230 If you start here you do[br]the green angle, then 0:06:22.230,0:06:23.570 you do the orange angle. 0:06:23.570,0:06:26.600 You go halfway around the[br]circle, and that'll give you, 0:06:26.600,0:06:28.720 it'll get you to 180 degrees. 0:06:28.720,0:06:32.870 So this green and orange angle[br]have to add up to 180 degrees 0:06:32.870,0:06:34.710 or they are supplementary. 0:06:34.710,0:06:37.120 And we've done other videos on[br]supplementary, but you just 0:06:37.120,0:06:40.720 have to realize they form the[br]same line or a half circle. 0:06:40.720,0:06:43.990 So if this right here is 70[br]degrees, then this orange angle 0:06:43.990,0:06:50.720 right here is 110 degrees,[br]because they add up to 180. 0:06:50.720,0:06:54.320 Now, if this character right[br]here is 110 degrees, what 0:06:54.320,0:06:56.660 do we know about this[br]character right here? 0:06:56.660,0:06:59.370 Well, this character is[br]opposite or vertical 0:06:59.370,0:07:02.450 to the 110 degrees so[br]it's also 110 degrees. 0:07:02.450,0:07:06.370 We also know since this angle[br]corresponds with this angle, 0:07:06.370,0:07:09.360 this angle will also[br]be 110 degrees. 0:07:09.360,0:07:11.830 Or we could have said that[br]look, because this is 70 and 0:07:11.830,0:07:14.180 this guy is supplementary,[br]these guys have to add up to 0:07:14.180,0:07:16.180 180 so you could have[br]gotten it that way. 0:07:16.180,0:07:19.270 And you could also figure out[br]that since this is 110, this 0:07:19.270,0:07:22.300 is a corresponding angle,[br]it is also going to be 110. 0:07:22.300,0:07:25.190 Or you could have said[br]this is opposite to 0:07:25.190,0:07:26.430 that so they're equal. 0:07:26.430,0:07:30.800 Or you could have said that[br]this is supplementary with 0:07:30.800,0:07:34.150 that angle, so 70 plus[br]110 have to be 180. 0:07:34.150,0:07:38.600 Or you could have said 70[br]plus this angle are 180. 0:07:38.600,0:07:41.810 So there's a bunch of ways[br]to come to figure out 0:07:41.810,0:07:43.740 which angle is which. 0:07:43.740,0:07:46.000 In the next video I'm just[br]going to do a bunch of examples 0:07:46.000,0:07:48.990 just to show that if you know[br]one of these angles, you 0:07:48.990,0:07:51.880 can really figure out[br]all of the angles. 0:07:51.880,0:07:52.242