WEBVTT 00:00:00.490 --> 00:00:02.322 In this video we're going to think a little bit about 00:00:02.322 --> 00:00:11.580 parallel lines, and other lines that intersect the parallel 00:00:11.580 --> 00:00:13.780 lines, and we call those transversals. 00:00:13.780 --> 00:00:16.810 So first let's think about what a parallel or what 00:00:16.810 --> 00:00:18.490 parallel lines are. 00:00:18.490 --> 00:00:21.700 So one definition we could use, and I think that'll work well 00:00:21.700 --> 00:00:24.220 for the purposes of this video, are they're two lines that 00:00:24.220 --> 00:00:25.660 sit in the same plane. 00:00:25.660 --> 00:00:29.090 And when I talk about a plane, I'm talking about a, you can 00:00:29.090 --> 00:00:32.490 imagine a flat two-dimensional surface like this screen -- 00:00:32.490 --> 00:00:33.910 this screen is a plane. 00:00:33.910 --> 00:00:37.730 So two lines that sit in a plane that never intersect. 00:00:37.730 --> 00:00:41.570 So this line -- I'll try my best to draw it -- and imagine 00:00:41.570 --> 00:00:43.750 the line just keeps going in that direction and that 00:00:43.750 --> 00:00:47.280 direction -- let me do another one in a different color -- 00:00:47.280 --> 00:00:52.050 and this line right here are parallel. 00:00:52.050 --> 00:00:53.690 They will never intersect. 00:00:53.690 --> 00:00:55.660 If you assume that I drew it straight enough and that 00:00:55.660 --> 00:00:58.000 they're going in the exact same direction, they 00:00:58.000 --> 00:00:59.840 will never intersect. 00:00:59.840 --> 00:01:02.070 And so if you think about what types of lines are not 00:01:02.070 --> 00:01:07.840 parallel, well, this green line and this pink line 00:01:07.840 --> 00:01:08.940 are not parallel. 00:01:08.940 --> 00:01:11.940 They clearly intersect at some point. 00:01:11.940 --> 00:01:15.350 So these two guys are parallel right over here, and sometimes 00:01:15.350 --> 00:01:18.690 it's specified, sometimes people will draw an arrow going 00:01:18.690 --> 00:01:20.900 in the same direction to show that those two lines 00:01:20.900 --> 00:01:21.840 are parallel. 00:01:21.840 --> 00:01:24.400 If there are multiple parallel lines, they might do two arrows 00:01:24.400 --> 00:01:25.760 and two arrows or whatever. 00:01:25.760 --> 00:01:27.270 But you just have to say OK, these lines will 00:01:27.270 --> 00:01:28.550 never intersect. 00:01:28.550 --> 00:01:31.060 What we want to think about is what happens when 00:01:31.060 --> 00:01:36.200 these parallel lines are intersected by a third line. 00:01:36.200 --> 00:01:37.820 Let me draw the third line here. 00:01:37.820 --> 00:01:41.690 So third line like this. 00:01:41.690 --> 00:01:45.970 And we call that, right there, the third line that intersects 00:01:45.970 --> 00:01:52.170 the parallel lines we call a transversal line. 00:01:52.170 --> 00:01:56.150 Because it tranverses the two parallel lines. 00:01:56.150 --> 00:01:59.230 Now whenever you have a transversal crossing parallel 00:01:59.230 --> 00:02:02.190 lines, you have an interesting relationship between 00:02:02.190 --> 00:02:03.320 the angles form. 00:02:03.320 --> 00:02:05.660 Now this shows up on a lot of standardized tests. 00:02:05.660 --> 00:02:09.200 It's kind of a core type of a geometry problem. 00:02:09.200 --> 00:02:12.450 So it's a good thing to really get clear in our heads. 00:02:12.450 --> 00:02:15.620 So the first thing to realize is if these lines are parallel, 00:02:15.620 --> 00:02:18.350 we're going to assume these lines are parallel, then we 00:02:18.350 --> 00:02:21.760 have corresponding angles are going to be the same. 00:02:21.760 --> 00:02:25.820 What I mean by corresponding angles are I guess you could 00:02:25.820 --> 00:02:28.840 think there are four angles that get formed when this 00:02:28.840 --> 00:02:31.195 purple line or this magenta line intersects 00:02:31.195 --> 00:02:32.350 this yellow line. 00:02:32.350 --> 00:02:38.070 You have this angle up here that I've specified in green, 00:02:38.070 --> 00:02:42.970 you have -- let me do another one in orange -- you have this 00:02:42.970 --> 00:02:48.280 angle right here in orange, you have this angle right here in 00:02:48.280 --> 00:02:52.600 this other shade of green, and then you have this angle 00:02:52.600 --> 00:02:55.290 right here -- right there that I've made in that 00:02:55.290 --> 00:02:56.930 bluish-purplish color. 00:02:56.930 --> 00:02:58.790 So those are the four angles. 00:02:58.790 --> 00:03:01.680 So when we talk about corresponding angles, we're 00:03:01.680 --> 00:03:04.770 talking about, for example, this top right angle in green 00:03:04.770 --> 00:03:08.930 up here, that corresponds to this top right angle in, what 00:03:08.930 --> 00:03:12.040 I can draw it in that same green, right over here. 00:03:12.040 --> 00:03:14.570 These two angles are corresponding. 00:03:14.570 --> 00:03:17.990 These two are corresponding angles and they're 00:03:17.990 --> 00:03:19.520 going to be equal. 00:03:19.520 --> 00:03:20.820 These are equal angles. 00:03:20.820 --> 00:03:24.410 If this is -- I'll make up a number -- if this is 70 00:03:24.410 --> 00:03:27.880 degrees, then this angle right here is also 00:03:27.880 --> 00:03:29.410 going to be 70 degrees. 00:03:29.410 --> 00:03:32.000 And if you just think about it, or if you even play with 00:03:32.000 --> 00:03:35.150 toothpicks or something, and you keep changing the direction 00:03:35.150 --> 00:03:38.140 of this transversal line, you'll see that it actually 00:03:38.140 --> 00:03:40.750 looks like they should always be equal. 00:03:40.750 --> 00:03:43.200 If I were to take -- let me draw two other parallel 00:03:43.200 --> 00:03:45.980 lines, let me show maybe a more extreme example. 00:03:45.980 --> 00:03:50.350 So if I have two other parallel lines like that, and then let 00:03:50.350 --> 00:03:57.340 me make a transversal that forms a smaller -- it's even a 00:03:57.340 --> 00:03:59.930 smaller angle here -- you see that this angle right here 00:03:59.930 --> 00:04:02.070 looks the same as that angle. 00:04:02.070 --> 00:04:05.340 Those are corresponding angles and they will be equivalent. 00:04:05.340 --> 00:04:08.330 From this perspective it's kind of the top right angle and each 00:04:08.330 --> 00:04:10.430 intersection is the same. 00:04:10.430 --> 00:04:13.600 Now the same is true of the other corresponding angles. 00:04:13.600 --> 00:04:16.660 This angle right here in this example, it's the top left 00:04:16.660 --> 00:04:21.120 angle will be the same as the top left angle right over here. 00:04:21.120 --> 00:04:27.080 This bottom left angle will be the same down here. 00:04:27.080 --> 00:04:30.000 If this right here is 70 degrees, then this down here 00:04:30.000 --> 00:04:32.040 will also be 70 degrees. 00:04:32.040 --> 00:04:36.040 And then finally, of course, this angle and this angle 00:04:36.040 --> 00:04:37.990 will also be the same. 00:04:37.990 --> 00:04:41.520 So corresponding angles -- let me write these -- these are 00:04:41.520 --> 00:04:43.170 corresponding angles are congruent. 00:04:46.640 --> 00:04:55.180 Corresponding angles are equal. 00:04:55.180 --> 00:04:57.050 And that and that are corresponding, that and 00:04:57.050 --> 00:04:59.400 that, that and that, and that and that. 00:04:59.400 --> 00:05:04.600 Now, the next set of equal angles to realize are sometimes 00:05:04.600 --> 00:05:06.610 they're called vertical angles, sometimes they're called 00:05:06.610 --> 00:05:08.440 opposite angles. 00:05:08.440 --> 00:05:11.700 But if you take this angle right here, the angle that is 00:05:11.700 --> 00:05:15.060 vertical to it or is opposite as you go right across the 00:05:15.060 --> 00:05:18.650 point of intersection is this angle right here, and that is 00:05:18.650 --> 00:05:20.580 going to be the same thing. 00:05:20.580 --> 00:05:23.860 So we could say opposite -- I like opposite because it's not 00:05:23.860 --> 00:05:25.720 always in the vertical direction, sometimes it's in 00:05:25.720 --> 00:05:27.650 the horizontal direction, but sometimes they're referred 00:05:27.650 --> 00:05:29.400 to as vertical angles. 00:05:29.400 --> 00:05:37.370 Opposite or vertical angles are also equal. 00:05:37.370 --> 00:05:40.940 So if that's 70 degrees, then this is also 70 degrees. 00:05:40.940 --> 00:05:43.980 And if this is 70 degrees, then this right here 00:05:43.980 --> 00:05:46.710 is also 70 degrees. 00:05:46.710 --> 00:05:49.240 So it's interesting, if that's 70 degrees and that's 70 00:05:49.240 --> 00:05:52.230 degrees, and if this is 70 degrees and that is also 70 00:05:52.230 --> 00:05:55.750 degrees, so no matter what this is, this will also be the same 00:05:55.750 --> 00:05:58.060 thing because this is the same as that, that 00:05:58.060 --> 00:05:59.770 is the same as that. 00:05:59.770 --> 00:06:07.180 Now, the last one that you need to I guess kind of realize are 00:06:07.180 --> 00:06:09.870 the relationship between this orange angle and this 00:06:09.870 --> 00:06:11.860 green angle right there. 00:06:11.860 --> 00:06:17.890 You can see that when you add up the angles, you go halfway 00:06:17.890 --> 00:06:19.710 around a circle, right? 00:06:19.710 --> 00:06:22.230 If you start here you do the green angle, then 00:06:22.230 --> 00:06:23.570 you do the orange angle. 00:06:23.570 --> 00:06:26.600 You go halfway around the circle, and that'll give you, 00:06:26.600 --> 00:06:28.720 it'll get you to 180 degrees. 00:06:28.720 --> 00:06:32.870 So this green and orange angle have to add up to 180 degrees 00:06:32.870 --> 00:06:34.710 or they are supplementary. 00:06:34.710 --> 00:06:37.120 And we've done other videos on supplementary, but you just 00:06:37.120 --> 00:06:40.720 have to realize they form the same line or a half circle. 00:06:40.720 --> 00:06:43.990 So if this right here is 70 degrees, then this orange angle 00:06:43.990 --> 00:06:50.720 right here is 110 degrees, because they add up to 180. 00:06:50.720 --> 00:06:54.320 Now, if this character right here is 110 degrees, what 00:06:54.320 --> 00:06:56.660 do we know about this character right here? 00:06:56.660 --> 00:06:59.370 Well, this character is opposite or vertical 00:06:59.370 --> 00:07:02.450 to the 110 degrees so it's also 110 degrees. 00:07:02.450 --> 00:07:06.370 We also know since this angle corresponds with this angle, 00:07:06.370 --> 00:07:09.360 this angle will also be 110 degrees. 00:07:09.360 --> 00:07:11.830 Or we could have said that look, because this is 70 and 00:07:11.830 --> 00:07:14.180 this guy is supplementary, these guys have to add up to 00:07:14.180 --> 00:07:16.180 180 so you could have gotten it that way. 00:07:16.180 --> 00:07:19.270 And you could also figure out that since this is 110, this 00:07:19.270 --> 00:07:22.300 is a corresponding angle, it is also going to be 110. 00:07:22.300 --> 00:07:25.190 Or you could have said this is opposite to 00:07:25.190 --> 00:07:26.430 that so they're equal. 00:07:26.430 --> 00:07:30.800 Or you could have said that this is supplementary with 00:07:30.800 --> 00:07:34.150 that angle, so 70 plus 110 have to be 180. 00:07:34.150 --> 00:07:38.600 Or you could have said 70 plus this angle are 180. 00:07:38.600 --> 00:07:41.810 So there's a bunch of ways to come to figure out 00:07:41.810 --> 00:07:43.740 which angle is which. 00:07:43.740 --> 00:07:46.000 In the next video I'm just going to do a bunch of examples 00:07:46.000 --> 00:07:48.990 just to show that if you know one of these angles, you 00:07:48.990 --> 00:07:51.880 can really figure out all of the angles.