[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.74,0:00:03.04,Default,,0000,0000,0000,,What we're going to\Nprove in this video
Dialogue: 0,0:00:03.04,0:00:05.30,Default,,0000,0000,0000,,is a couple of fairly\Nstraightforward
Dialogue: 0,0:00:05.30,0:00:06.84,Default,,0000,0000,0000,,parallelogram-related proofs.
Dialogue: 0,0:00:06.84,0:00:08.67,Default,,0000,0000,0000,,And this first one,\Nwe're going to say, hey,
Dialogue: 0,0:00:08.67,0:00:10.72,Default,,0000,0000,0000,,if we have this\Nparallelogram ABCD,
Dialogue: 0,0:00:10.72,0:00:13.92,Default,,0000,0000,0000,,let's prove that the opposite\Nsides have the same length.
Dialogue: 0,0:00:13.92,0:00:19.82,Default,,0000,0000,0000,,So prove that AB is equal to\NDC and that AD is equal to BC.
Dialogue: 0,0:00:19.82,0:00:21.63,Default,,0000,0000,0000,,So let me draw a diagonal here.
Dialogue: 0,0:00:24.74,0:00:26.85,Default,,0000,0000,0000,,And this diagonal, depending\Non how you view it,
Dialogue: 0,0:00:26.85,0:00:29.90,Default,,0000,0000,0000,,is intersecting two\Nsets of parallel lines.
Dialogue: 0,0:00:29.90,0:00:32.15,Default,,0000,0000,0000,,So you could also consider\Nit to be a transversal.
Dialogue: 0,0:00:32.15,0:00:34.44,Default,,0000,0000,0000,,Actually, let me draw it a\Nlittle bit neater than that.
Dialogue: 0,0:00:34.44,0:00:35.79,Default,,0000,0000,0000,,I can do a better job.
Dialogue: 0,0:00:35.79,0:00:36.29,Default,,0000,0000,0000,,Nope.
Dialogue: 0,0:00:36.29,0:00:38.55,Default,,0000,0000,0000,,That's not any better.
Dialogue: 0,0:00:38.55,0:00:41.53,Default,,0000,0000,0000,,That is about as\Ngood as I can do.
Dialogue: 0,0:00:41.53,0:00:44.00,Default,,0000,0000,0000,,So if we view DB,\Nthis diagonal DB--
Dialogue: 0,0:00:44.00,0:00:47.78,Default,,0000,0000,0000,,we can view it as a transversal\Nfor the parallel lines AB
Dialogue: 0,0:00:47.78,0:00:49.11,Default,,0000,0000,0000,,and DC.
Dialogue: 0,0:00:49.11,0:00:51.80,Default,,0000,0000,0000,,And if you view it that\Nway, you can pick out
Dialogue: 0,0:00:51.80,0:00:56.48,Default,,0000,0000,0000,,that angle ABD is going to\Nbe congruent-- so angle ABD.
Dialogue: 0,0:00:56.48,0:00:57.90,Default,,0000,0000,0000,,That's that angle\Nright there-- is
Dialogue: 0,0:00:57.90,0:01:00.96,Default,,0000,0000,0000,,going to be congruent\Nto angle BDC,
Dialogue: 0,0:01:00.96,0:01:03.43,Default,,0000,0000,0000,,because they are\Nalternate interior angles.
Dialogue: 0,0:01:03.43,0:01:05.30,Default,,0000,0000,0000,,You have a transversal--\Nparallel lines.
Dialogue: 0,0:01:05.30,0:01:09.81,Default,,0000,0000,0000,,So we know that\Nangle ABD is going
Dialogue: 0,0:01:09.81,0:01:12.01,Default,,0000,0000,0000,,to be congruent to angle BDC.
Dialogue: 0,0:01:16.02,0:01:18.36,Default,,0000,0000,0000,,Now, you could also\Nview this diagonal,
Dialogue: 0,0:01:18.36,0:01:21.49,Default,,0000,0000,0000,,DB-- you could view it as\Na transversal of these two
Dialogue: 0,0:01:21.49,0:01:26.03,Default,,0000,0000,0000,,parallel lines, of the other\Npair of parallel lines, AD
Dialogue: 0,0:01:26.03,0:01:27.60,Default,,0000,0000,0000,,and BC.
Dialogue: 0,0:01:27.60,0:01:29.99,Default,,0000,0000,0000,,And if you look at it that\Nway, then you immediately
Dialogue: 0,0:01:29.99,0:01:38.13,Default,,0000,0000,0000,,see that angle DBC\Nright over here
Dialogue: 0,0:01:38.13,0:01:40.49,Default,,0000,0000,0000,,is going to be\Ncongruent to angle
Dialogue: 0,0:01:40.49,0:01:47.69,Default,,0000,0000,0000,,ADB for the exact same reason.
Dialogue: 0,0:01:47.69,0:01:49.63,Default,,0000,0000,0000,,They are alternate\Ninterior angles
Dialogue: 0,0:01:49.63,0:01:53.17,Default,,0000,0000,0000,,of a transversal intersecting\Nthese two parallel lines.
Dialogue: 0,0:01:53.17,0:01:54.36,Default,,0000,0000,0000,,So I could write this.
Dialogue: 0,0:01:54.36,0:02:00.81,Default,,0000,0000,0000,,This is alternate\Ninterior angles
Dialogue: 0,0:02:00.81,0:02:04.73,Default,,0000,0000,0000,,are congruent when you have\Na transversal intersecting
Dialogue: 0,0:02:04.73,0:02:06.88,Default,,0000,0000,0000,,two parallel lines.
Dialogue: 0,0:02:06.88,0:02:09.52,Default,,0000,0000,0000,,And we also see that\Nboth of these triangles,
Dialogue: 0,0:02:09.52,0:02:15.91,Default,,0000,0000,0000,,triangle ADB and triangle CDB,\Nboth share this side over here.
Dialogue: 0,0:02:15.91,0:02:18.44,Default,,0000,0000,0000,,It's obviously equal to itself.
Dialogue: 0,0:02:18.44,0:02:20.42,Default,,0000,0000,0000,,Now, why is this useful?
Dialogue: 0,0:02:20.42,0:02:22.05,Default,,0000,0000,0000,,Well, you might\Nrealize that we've just
Dialogue: 0,0:02:22.05,0:02:23.82,Default,,0000,0000,0000,,shown that both of\Nthese triangles,
Dialogue: 0,0:02:23.82,0:02:25.34,Default,,0000,0000,0000,,they have this pink angle.
Dialogue: 0,0:02:25.34,0:02:27.53,Default,,0000,0000,0000,,Then they have this\Nside in common.
Dialogue: 0,0:02:27.53,0:02:28.99,Default,,0000,0000,0000,,And then they have\Nthe green angle.
Dialogue: 0,0:02:28.99,0:02:32.58,Default,,0000,0000,0000,,Pink angle, side in common,\Nand then the green angle.
Dialogue: 0,0:02:32.58,0:02:35.47,Default,,0000,0000,0000,,So we've just shown\Nby angle-side-angle
Dialogue: 0,0:02:35.47,0:02:38.04,Default,,0000,0000,0000,,that these two\Ntriangles are congruent.
Dialogue: 0,0:02:38.04,0:02:39.44,Default,,0000,0000,0000,,So let me write this down.
Dialogue: 0,0:02:39.44,0:02:42.32,Default,,0000,0000,0000,,We have shown that\Ntriangle-- I'll
Dialogue: 0,0:02:42.32,0:02:47.65,Default,,0000,0000,0000,,go from non-labeled\Nto pink to green-- ADB
Dialogue: 0,0:02:47.65,0:02:53.10,Default,,0000,0000,0000,,is congruent to triangle--\Nnon-labeled to pink to green--
Dialogue: 0,0:02:53.10,0:02:53.60,Default,,0000,0000,0000,,CBD.
Dialogue: 0,0:02:58.81,0:03:02.44,Default,,0000,0000,0000,,And this comes out of\Nangle-side-angle congruency.
Dialogue: 0,0:03:09.39,0:03:11.19,Default,,0000,0000,0000,,Well, what does that do for us?
Dialogue: 0,0:03:11.19,0:03:13.36,Default,,0000,0000,0000,,Well, if two triangles\Nare congruent, then
Dialogue: 0,0:03:13.36,0:03:16.35,Default,,0000,0000,0000,,all of the corresponding\Nfeatures of the two triangles
Dialogue: 0,0:03:16.35,0:03:18.01,Default,,0000,0000,0000,,are going to be congruent.
Dialogue: 0,0:03:18.01,0:03:26.36,Default,,0000,0000,0000,,In particular, side DC\Non this bottom triangle
Dialogue: 0,0:03:26.36,0:03:29.11,Default,,0000,0000,0000,,corresponds to side BA\Non that top triangle.
Dialogue: 0,0:03:29.11,0:03:32.62,Default,,0000,0000,0000,,So they need to be congruent.
Dialogue: 0,0:03:32.62,0:03:38.23,Default,,0000,0000,0000,,So we get DC is going\Nto be equal to BA.
Dialogue: 0,0:03:38.23,0:03:43.37,Default,,0000,0000,0000,,And that's because they\Nare corresponding sides
Dialogue: 0,0:03:43.37,0:03:47.20,Default,,0000,0000,0000,,of congruent triangles.
Dialogue: 0,0:03:47.20,0:03:49.56,Default,,0000,0000,0000,,So this is going to\Nbe equal to that.
Dialogue: 0,0:03:49.56,0:03:54.19,Default,,0000,0000,0000,,And by that exact same\Nlogic, AD corresponds to CB.
Dialogue: 0,0:03:58.26,0:04:00.77,Default,,0000,0000,0000,,AD is equal to CB.
Dialogue: 0,0:04:00.77,0:04:03.68,Default,,0000,0000,0000,,And for the exact same\Nreason-- corresponding sides
Dialogue: 0,0:04:03.68,0:04:05.24,Default,,0000,0000,0000,,of congruent triangles.
Dialogue: 0,0:04:05.24,0:04:06.74,Default,,0000,0000,0000,,And then we're done.
Dialogue: 0,0:04:06.74,0:04:09.90,Default,,0000,0000,0000,,We've proven that opposite\Nsides are congruent.
Dialogue: 0,0:04:09.90,0:04:11.02,Default,,0000,0000,0000,,Now let's go the other way.
Dialogue: 0,0:04:13.64,0:04:16.27,Default,,0000,0000,0000,,Let's say that we have some\Ntype of a quadrilateral,
Dialogue: 0,0:04:16.27,0:04:18.83,Default,,0000,0000,0000,,and we know that the\Nopposite sides are congruent.
Dialogue: 0,0:04:18.83,0:04:22.46,Default,,0000,0000,0000,,Can we prove to ourselves\Nthat this is a parallelogram?
Dialogue: 0,0:04:22.46,0:04:24.90,Default,,0000,0000,0000,,Well, it's kind of the\Nsame proof in reverse.
Dialogue: 0,0:04:24.90,0:04:27.08,Default,,0000,0000,0000,,So let's draw a\Ndiagonal here, since we
Dialogue: 0,0:04:27.08,0:04:29.25,Default,,0000,0000,0000,,know a lot about triangles.
Dialogue: 0,0:04:29.25,0:04:31.76,Default,,0000,0000,0000,,So let me draw.
Dialogue: 0,0:04:31.76,0:04:34.20,Default,,0000,0000,0000,,There we go.
Dialogue: 0,0:04:34.20,0:04:35.34,Default,,0000,0000,0000,,That's the hardest part.
Dialogue: 0,0:04:35.34,0:04:36.41,Default,,0000,0000,0000,,Draw it.
Dialogue: 0,0:04:36.41,0:04:37.93,Default,,0000,0000,0000,,That's pretty good.
Dialogue: 0,0:04:37.93,0:04:38.43,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:04:38.43,0:04:42.71,Default,,0000,0000,0000,,So we obviously know that CB\Nis going to be equal to itself.
Dialogue: 0,0:04:42.71,0:04:44.60,Default,,0000,0000,0000,,So I'll draw it like that.
Dialogue: 0,0:04:44.60,0:04:46.70,Default,,0000,0000,0000,,Obviously, because\Nit's the same line.
Dialogue: 0,0:04:46.70,0:04:48.32,Default,,0000,0000,0000,,And then we have\Nsomething interesting.
Dialogue: 0,0:04:48.32,0:04:51.50,Default,,0000,0000,0000,,We've split this quadrilateral\Ninto two triangles, triangle
Dialogue: 0,0:04:51.50,0:04:56.50,Default,,0000,0000,0000,,ACB and triangle DBC.
Dialogue: 0,0:04:56.50,0:05:00.43,Default,,0000,0000,0000,,And notice, all three sides\Nof these two triangles
Dialogue: 0,0:05:00.43,0:05:01.65,Default,,0000,0000,0000,,are equal to each other.
Dialogue: 0,0:05:01.65,0:05:05.11,Default,,0000,0000,0000,,So we know by side-side-side\Nthat they are congruent.
Dialogue: 0,0:05:05.11,0:05:11.12,Default,,0000,0000,0000,,So we know that triangle\NA-- and we're starting at A,
Dialogue: 0,0:05:11.12,0:05:13.38,Default,,0000,0000,0000,,and then I'm going\Nto the one-hash side.
Dialogue: 0,0:05:13.38,0:05:20.26,Default,,0000,0000,0000,,So ACB is congruent\Nto triangle DBC.
Dialogue: 0,0:05:24.03,0:05:30.67,Default,,0000,0000,0000,,And this is by\Nside-side-side congruency.
Dialogue: 0,0:05:30.67,0:05:32.30,Default,,0000,0000,0000,,Well, what does that do for us?
Dialogue: 0,0:05:32.30,0:05:34.55,Default,,0000,0000,0000,,Well, it tells us that all\Nof the corresponding angles
Dialogue: 0,0:05:34.55,0:05:36.43,Default,,0000,0000,0000,,are going to be congruent.
Dialogue: 0,0:05:36.43,0:05:42.20,Default,,0000,0000,0000,,So for example, angle ABC\Nis going to be-- so let
Dialogue: 0,0:05:42.20,0:05:43.83,Default,,0000,0000,0000,,me mark that.
Dialogue: 0,0:05:49.16,0:05:51.78,Default,,0000,0000,0000,,You can say ABC is going\Nto be congruent to DCB.
Dialogue: 0,0:05:57.48,0:06:04.30,Default,,0000,0000,0000,,And you could say, by\Ncorresponding angles congruent
Dialogue: 0,0:06:04.30,0:06:06.56,Default,,0000,0000,0000,,of congruent triangles.
Dialogue: 0,0:06:06.56,0:06:09.20,Default,,0000,0000,0000,,I'm just using some shorthand\Nhere to save some time.
Dialogue: 0,0:06:09.20,0:06:12.08,Default,,0000,0000,0000,,So ABC is going to\Nbe congruent to DCB,
Dialogue: 0,0:06:12.08,0:06:15.54,Default,,0000,0000,0000,,so these two angles are\Ngoing to be congruent.
Dialogue: 0,0:06:15.54,0:06:18.36,Default,,0000,0000,0000,,Well, this is interesting,\Nbecause here you have a line.
Dialogue: 0,0:06:18.36,0:06:21.46,Default,,0000,0000,0000,,And it's intersecting AB and CD.
Dialogue: 0,0:06:21.46,0:06:23.97,Default,,0000,0000,0000,,And we clearly see\Nthat these things that
Dialogue: 0,0:06:23.97,0:06:27.72,Default,,0000,0000,0000,,could be alternate interior\Nangles are congruent.
Dialogue: 0,0:06:27.72,0:06:30.36,Default,,0000,0000,0000,,And because we have these\Ncongruent alternate interior
Dialogue: 0,0:06:30.36,0:06:34.20,Default,,0000,0000,0000,,angles, we know that AB\Nmust be parallel to CD.
Dialogue: 0,0:06:34.20,0:06:36.71,Default,,0000,0000,0000,,So this must be\Nparallel to that.
Dialogue: 0,0:06:36.71,0:06:41.63,Default,,0000,0000,0000,,So we know that AB\Nis parallel to CD
Dialogue: 0,0:06:41.63,0:06:50.35,Default,,0000,0000,0000,,by alternate interior angles\Nof a transversal intersecting
Dialogue: 0,0:06:50.35,0:06:51.73,Default,,0000,0000,0000,,parallel lines.
Dialogue: 0,0:06:51.73,0:06:53.84,Default,,0000,0000,0000,,Now, we can use that\Nexact same logic.
Dialogue: 0,0:06:53.84,0:06:56.72,Default,,0000,0000,0000,,We also know that angle--\Nlet me get this right.
Dialogue: 0,0:06:56.72,0:07:03.78,Default,,0000,0000,0000,,Angle ACB is congruent\Nto angle DBC.
Dialogue: 0,0:07:09.71,0:07:16.17,Default,,0000,0000,0000,,And we know that by\Ncorresponding angles congruent
Dialogue: 0,0:07:16.17,0:07:18.79,Default,,0000,0000,0000,,of congruent triangles.
Dialogue: 0,0:07:18.79,0:07:22.58,Default,,0000,0000,0000,,So we're just saying this\Nangle is equal to that angle.
Dialogue: 0,0:07:22.58,0:07:25.05,Default,,0000,0000,0000,,Well, once again, these could\Nbe alternate interior angles.
Dialogue: 0,0:07:25.05,0:07:26.26,Default,,0000,0000,0000,,They look like they could be.
Dialogue: 0,0:07:26.26,0:07:27.28,Default,,0000,0000,0000,,This is a transversal.
Dialogue: 0,0:07:27.28,0:07:28.98,Default,,0000,0000,0000,,And here's two lines\Nhere, which we're not sure
Dialogue: 0,0:07:28.98,0:07:30.10,Default,,0000,0000,0000,,whether they're parallel.
Dialogue: 0,0:07:30.10,0:07:33.03,Default,,0000,0000,0000,,But because the alternate\Ninterior angles are congruent,
Dialogue: 0,0:07:33.03,0:07:34.99,Default,,0000,0000,0000,,we know that they are parallel.
Dialogue: 0,0:07:34.99,0:07:36.86,Default,,0000,0000,0000,,So this is parallel to that.
Dialogue: 0,0:07:36.86,0:07:41.50,Default,,0000,0000,0000,,So we know that AC\Nis parallel to BD
Dialogue: 0,0:07:41.50,0:07:43.96,Default,,0000,0000,0000,,by alternate interior angles.
Dialogue: 0,0:07:48.66,0:07:49.61,Default,,0000,0000,0000,,And we're done.
Dialogue: 0,0:07:49.61,0:07:51.39,Default,,0000,0000,0000,,So what we've done\Nis-- it's interesting.
Dialogue: 0,0:07:51.39,0:07:55.68,Default,,0000,0000,0000,,We've shown if you\Nhave a parallelogram,
Dialogue: 0,0:07:55.68,0:07:57.64,Default,,0000,0000,0000,,opposite sides have\Nthe same length.
Dialogue: 0,0:07:57.64,0:07:59.67,Default,,0000,0000,0000,,And if opposite sides\Nhave the same length,
Dialogue: 0,0:07:59.67,0:08:01.03,Default,,0000,0000,0000,,then you have a parallelogram.
Dialogue: 0,0:08:01.03,0:08:03.34,Default,,0000,0000,0000,,And so we've actually proven\Nit in both directions.
Dialogue: 0,0:08:03.34,0:08:06.08,Default,,0000,0000,0000,,And so we can actually make what\Nyou call an "if and only if"
Dialogue: 0,0:08:06.08,0:08:06.58,Default,,0000,0000,0000,,statement.
Dialogue: 0,0:08:12.15,0:08:16.02,Default,,0000,0000,0000,,You could say opposite sides of\Na quadrilateral are parallel if
Dialogue: 0,0:08:16.02,0:08:18.52,Default,,0000,0000,0000,,and only if their\Nlengths are equal.
Dialogue: 0,0:08:18.52,0:08:20.08,Default,,0000,0000,0000,,And you say if and only if.
Dialogue: 0,0:08:20.08,0:08:21.62,Default,,0000,0000,0000,,So if they are\Nparallel, then you
Dialogue: 0,0:08:21.62,0:08:23.04,Default,,0000,0000,0000,,could say their\Nlengths are equal.
Dialogue: 0,0:08:23.04,0:08:26.41,Default,,0000,0000,0000,,And only if their lengths\Nare equal are they parallel.
Dialogue: 0,0:08:26.41,0:08:28.88,Default,,0000,0000,0000,,We've proven it in\Nboth directions.