WEBVTT 00:00:00.720 --> 00:00:02.550 So, we have a parallelogram right over here 00:00:02.560 --> 00:00:06.660 So, what we wanna prove is that it's diagonals bisect each other 00:00:06.670 --> 00:00:10.040 So, the first thing we can think about; these aren't just diagonals, 00:00:10.050 --> 00:00:12.460 these are lines that are intersecting parallel lines 00:00:12.470 --> 00:00:14.560 So, you can also view them as transversals 00:00:14.570 --> 00:00:19.540 And if we focus on DB right over here, we see that it intersects DC 00:00:19.550 --> 00:00:21.890 and AB and it sits there 00:00:21.900 --> 00:00:23.640 Those we know are parallelograms 00:00:23.650 --> 00:00:24.960 We know that they are parallel 00:00:24.970 --> 00:00:25.990 This is a parallelogram 00:00:26.000 --> 00:00:28.640 Alternate interior angles must be congruent 00:00:28.650 --> 00:00:31.360 So, that angle must be equal to that angle there 00:00:31.370 --> 00:00:32.670 Let me make a label here 00:00:32.680 --> 00:00:34.030 Let me call that middle point E 00:00:34.040 --> 00:00:42.630 So, we know that angle ABE must be congruent to angle CDE 00:00:42.640 --> 00:00:50.130 by alternate interior angles of 00:00:50.140 --> 00:00:52.130 a transversal intersecting parallel line 00:00:52.140 --> 00:00:56.680 Alternate interior angles 00:00:56.690 --> 00:01:00.840 If we look at diagonal AC or we should call it transversal AC 00:01:00.850 --> 00:01:02.520 we can make the same argument 00:01:02.730 --> 00:01:04.470 It intersects here and here 00:01:04.480 --> 00:01:06.220 These two lines are parallel 00:01:06.230 --> 00:01:09.360 So, alternate interior angles must congruent 00:01:09.370 --> 00:01:12.740 So, angle DEC must be --- let me write this down --- 00:01:12.750 --> 00:01:19.050 angle DEC must be congruent to angle BAE 00:01:24.780 --> 00:01:27.150 by for the exact same reason 00:01:27.160 --> 00:01:28.680 Now we have something interesting 00:01:28.690 --> 00:01:31.580 If we look this top triangle over here and this bottom triangle, 00:01:31.590 --> 00:01:34.820 we have one set of corresponding angles that are congruent 00:01:34.830 --> 00:01:39.610 We have a side in between that's going to be congruent 00:01:39.620 --> 00:01:41.220 Actually, let me write that down explicitly 00:01:41.230 --> 00:01:46.380 We know and we've proved this to ourselves in the previous video 00:01:46.670 --> 00:01:50.380 that parallelograms not only are opposite sides are parallel they're 00:01:50.390 --> 00:01:51.540 also congruent 00:01:51.550 --> 00:01:54.310 So, we know from the previous video that that side is equal 00:01:54.320 --> 00:01:55.230 to that side 00:01:55.240 --> 00:01:56.840 So, let me go back to what I was saying 00:01:56.850 --> 00:01:59.760 We have two sets of corresponding angles that are congruent 00:01:59.770 --> 00:02:02.710 We have a side in between that's congruent 00:02:02.720 --> 00:02:04.740 And then we have another set of corresponding angles 00:02:04.750 --> 00:02:05.770 that are congruent 00:02:05.780 --> 00:02:08.150 So, we know that this triangle is congruent to that triangle 00:02:08.160 --> 00:02:10.320 by angle-side-angle 00:02:11.810 --> 00:02:15.960 So, we know that triangle --- I'm gonna go from the blue 00:02:15.970 --> 00:02:17.460 to the orange to the last one 00:02:17.470 --> 00:02:23.120 Triangle ABE is congruent to triangle blue, orange 00:02:23.130 --> 00:02:29.970 and the last one, CDE by angle-side-angle congruency 00:02:33.720 --> 00:02:35.940 Now what is that do for us 00:02:35.950 --> 00:02:38.860 What we know if two triangles are congruent, all of their 00:02:38.870 --> 00:02:41.370 corresponding features especially all of the corresponding 00:02:41.380 --> 00:02:42.620 sides are congruent 00:02:42.630 --> 00:02:47.740 So, we know that side EC corresponds to EA 00:02:47.750 --> 00:02:51.920 Or I could say side AE, we could say side AE, 00:02:55.240 --> 00:02:59.470 corresponds to side CE 00:03:00.990 --> 00:03:02.830 They're corresponding sides of congruent triangle 00:03:02.840 --> 00:03:05.360 So, their measures or their lengths must be the same 00:03:05.370 --> 00:03:08.850 So, AE must be equal to CE 00:03:08.860 --> 00:03:12.320 Let me put two slashes since I already used one slash over here 00:03:18.210 --> 00:03:24.320 let me focus on this -- we know that BE must be equal to DE 00:03:25.950 --> 00:03:29.450 Once again they're corresponding sides of two congruent triangles 00:03:29.460 --> 00:03:30.870 so they must have the same length 00:03:30.880 --> 00:03:38.320 So, this is corresponding sides of congruent triangles 00:03:38.330 --> 00:03:43.000 So, BE is equal to DE 00:03:43.010 --> 00:03:44.080 And we've done our proof 00:03:44.090 --> 00:03:48.780 We've showed that, look, diagonal DB is splitting AC into two 00:03:48.790 --> 00:03:51.230 segments of equal length and vice versa 00:03:51.240 --> 00:03:55.780 AC is splitting DB into two segments of equal lengths 00:03:55.790 --> 00:03:58.070 So, they are bisecting each other 00:03:58.080 --> 00:03:59.640 Now, let's go the other way around 00:03:59.650 --> 00:04:03.920 Let's prove to ourselves that if we have two diagonals 00:04:03.930 --> 00:04:06.980 of a quadrilateral that are bisecting each other that we're 00:04:06.990 --> 00:04:08.810 dealing with a parallelogram 00:04:08.820 --> 00:04:10.020 So, let me see 00:04:10.030 --> 00:04:12.010 So, we're gonna assume that the two diagonals 00:04:12.020 --> 00:04:13.150 are bisecting each other 00:04:13.160 --> 00:04:14.980 So, we're assuming that that is equal to that 00:04:14.990 --> 00:04:17.360 And that that, right over there, is equal to that 00:04:17.370 --> 00:04:22.290 Given that we wanna prove that this is a parallelogram 00:04:22.300 --> 00:04:25.160 And to do that we just have to remind ourselves 00:04:25.440 --> 00:04:30.000 We just have to remind ourselves that this angle is going 00:04:30.010 --> 00:04:31.040 to be equal to that angle 00:04:31.050 --> 00:04:33.730 One of the first things we learn because they're vertical angles 00:04:33.740 --> 00:04:34.640 So, let me write this down 00:04:34.650 --> 00:04:43.580 C -- label this point -- angle CED is going to be equal to 00:04:43.590 --> 00:04:52.390 or is congruent to angle, so I started is BEA, angle BEA 00:04:52.400 --> 00:04:55.200 And that, what is that, well that shows us that these 00:04:55.210 --> 00:04:57.810 two triangles are congruent 'cause we have a corresponding sides 00:04:57.820 --> 00:05:00.310 of a congruent and angle in between and on the other side 00:05:00.320 --> 00:05:03.810 So, we now know that the triangle, I'll keep this in yellow, 00:05:03.820 --> 00:05:20.300 triangle AEB is congruent to triangle DEC by side-angle-side 00:05:20.310 --> 00:05:28.170 congruency, by SAS congruent triangles 00:05:28.180 --> 00:05:29.160 Fair enough 00:05:29.170 --> 00:05:31.760 Now, if we know that two triangles congruent we know that all 00:05:31.770 --> 00:05:34.220 corresponding sides and angles are congruent 00:05:34.230 --> 00:05:44.580 So, for example, we know that angle CDE is going to be congruent 00:05:44.590 --> 00:05:48.360 to angle BAE 00:05:55.650 --> 00:06:05.790 And this is just corresponding angles of congruent triangles 00:06:05.800 --> 00:06:12.430 And now we have this kind of transversal of these two lines that 00:06:12.440 --> 00:06:16.570 could be parallel if the alternate interior angles are congruent 00:06:16.580 --> 00:06:17.990 And we see that they are 00:06:18.000 --> 00:06:22.470 These two are kind of candidate alternate interior angles and 00:06:22.480 --> 00:06:23.910 they are congruent 00:06:23.920 --> 00:06:26.870 So, AB must be parallel to CD 00:06:26.880 --> 00:06:31.780 So, AB, let's just draw one arrow, AB must be parallel to CD 00:06:34.950 --> 00:06:42.620 by alternate interior angles congruent of parallel lines 00:06:42.800 --> 00:06:46.110 I'm just writing in some short hand, forgive the cryptic nature 00:06:46.120 --> 00:06:47.670 of it although I'm saying it out 00:06:47.680 --> 00:06:50.300 And so we can then do the exact same -- while we just shown 00:06:50.310 --> 00:06:53.230 that these two sides are parallel -- we can do that exact same 00:06:53.240 --> 00:06:55.640 logic to show that these two sides are parallel 00:06:55.650 --> 00:06:57.090 I won't necessarily write it all out 00:06:57.100 --> 00:06:59.970 It's exact same proof to show that these two 00:06:59.980 --> 00:07:03.680 So, first of all, we know that this angle is congruent to that angle 00:07:03.690 --> 00:07:04.630 right over there 00:07:04.640 --> 00:07:06.930 And then we know, actually let me write it out, we know 00:07:06.940 --> 00:07:18.670 that angle AEC is congruent to angle DEB, I should say 00:07:22.650 --> 00:07:24.360 They are vertical angles 00:07:26.980 --> 00:07:29.060 And that is the reason up here as well 00:07:29.070 --> 00:07:31.920 Vertical angles 00:07:31.930 --> 00:07:35.260 And then we see that triangle AEC must be congruent 00:07:35.270 --> 00:07:38.270 to triangle DEB by side-angle-side 00:07:38.600 --> 00:07:45.010 So, then we have triangle AEC must be congruent to triangle 00:07:45.020 --> 00:07:50.890 DEB by SAScongruency 00:07:50.900 --> 00:07:53.730 Now, we know that corresponding angles must be congruent 00:07:53.740 --> 00:07:58.680 So, that we know that angle, so, for example angle CAE 00:08:01.760 --> 00:08:10.970 must be congruent to angle BDE and this is the corresponding 00:08:10.980 --> 00:08:13.510 angles of congruent triangles 00:08:13.520 --> 00:08:17.950 So, CAE, let me use a new color 00:08:18.130 --> 00:08:25.940 CAE must be congruent to BDE 00:08:28.050 --> 00:08:30.100 And now we have a transversal 00:08:30.110 --> 00:08:32.100 The alternate interior angles are congruent 00:08:32.110 --> 00:08:34.690 So, the two lines that the transversals are intersecting 00:08:34.700 --> 00:08:36.130 must be parallel 00:08:36.140 --> 00:08:39.230 So, this must be parallel to that 00:08:39.240 --> 00:08:44.440 So, then we have AC must be parallel to BD 00:08:45.490 --> 00:08:47.970 by alternate interior angles 00:08:50.560 --> 00:08:51.360 And we're done 00:08:51.370 --> 00:08:53.970 We've just proven that if the diagonals bisect each other, 00:08:53.980 --> 00:08:57.910 if we start that as a given then we end at a point where we say, 00:08:57.920 --> 00:09:00.860 "Hey, the opposite sides of this quadrilateral must be parallel 00:09:00.870 --> 00:09:04.690 or that ABCD is a parallelogram "