[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:05.00,Default,,0000,0000,0000,,Quadrilateral ABCD is a rhombus
Dialogue: 0,0:00:05.01,0:00:08.76,Default,,0000,0000,0000,,What they want us to prove is that their diagonals are perpendicular,
Dialogue: 0,0:00:08.77,0:00:11.58,Default,,0000,0000,0000,,that AC is perpendicular to BD
Dialogue: 0,0:00:11.59,0:00:14.17,Default,,0000,0000,0000,,Let's think about everything we know about a rhombus
Dialogue: 0,0:00:14.18,0:00:17.40,Default,,0000,0000,0000,,First of all, a rhombus is a special case of a parallelogram
Dialogue: 0,0:00:17.41,0:00:20.77,Default,,0000,0000,0000,,In a parallelogram, the opposite sides are parallel
Dialogue: 0,0:00:20.78,0:00:22.39,Default,,0000,0000,0000,,That side is parallel to that side
Dialogue: 0,0:00:22.40,0:00:24.90,Default,,0000,0000,0000,,These 2 sides are parallel
Dialogue: 0,0:00:24.91,0:00:28.09,Default,,0000,0000,0000,,In a rhombus, not only are the opposite sides parallel,
Dialogue: 0,0:00:28.10,0:00:31.69,Default,,0000,0000,0000,,but also all the sides have equal length
Dialogue: 0,0:00:31.70,0:00:36.56,Default,,0000,0000,0000,,This side is equal to this side, which is equal to that side,
Dialogue: 0,0:00:36.57,0:00:38.96,Default,,0000,0000,0000,,which is equal to that side right over there
Dialogue: 0,0:00:38.97,0:00:41.61,Default,,0000,0000,0000,,There's other interesting things we know about
Dialogue: 0,0:00:41.62,0:00:43.78,Default,,0000,0000,0000,,the diagonals of a parallelogram,
Dialogue: 0,0:00:43.79,0:00:46.09,Default,,0000,0000,0000,,which we know all rhombi are parallelograms
Dialogue: 0,0:00:46.10,0:00:48.72,Default,,0000,0000,0000,,The other way around is not necessarily true
Dialogue: 0,0:00:48.73,0:00:52.56,Default,,0000,0000,0000,,We know that for any parallelogram, and a rhombus is a parallelogram,
Dialogue: 0,0:00:52.57,0:00:55.59,Default,,0000,0000,0000,,that the diagonals bisect each other
Dialogue: 0,0:00:55.60,0:00:59.64,Default,,0000,0000,0000,,For example, let me label this point in the center, point E
Dialogue: 0,0:00:59.65,0:01:04.39,Default,,0000,0000,0000,,We know that AE is going to be equal to EC,
Dialogue: 0,0:01:04.40,0:01:06.25,Default,,0000,0000,0000,,I'll put 2 slashes right over there
Dialogue: 0,0:01:06.26,0:01:13.35,Default,,0000,0000,0000,,We also know that EB is going to be equal to ED
Dialogue: 0,0:01:13.36,0:01:18.84,Default,,0000,0000,0000,,This is all of what we know, when someone just says that ABCD is a
Dialogue: 0,0:01:18.85,0:01:22.04,Default,,0000,0000,0000,,rhombus, based on other things that we've proven to ourselves
Dialogue: 0,0:01:22.05,0:01:26.84,Default,,0000,0000,0000,,Now we're gonna prove that AC is perpendicular to BD
Dialogue: 0,0:01:26.85,0:01:29.01,Default,,0000,0000,0000,,An interesting way to prove it,
Dialogue: 0,0:01:29.02,0:01:30.93,Default,,0000,0000,0000,,and you can look at it just by eyeballing it,
Dialogue: 0,0:01:30.94,0:01:34.51,Default,,0000,0000,0000,,is if we can show that this triangle is congruent to this triangle
Dialogue: 0,0:01:34.52,0:01:38.36,Default,,0000,0000,0000,,and that these 2 angles right over here correspond to each other
Dialogue: 0,0:01:38.37,0:01:40.84,Default,,0000,0000,0000,,then they have to be the same and they'll be supplementary
Dialogue: 0,0:01:40.85,0:01:43.27,Default,,0000,0000,0000,,and they'll be 90 degrees so let's just prove it to ourselves
Dialogue: 0,0:01:43.28,0:01:47.94,Default,,0000,0000,0000,,The first thing we see is we have a side, a side, and a side
Dialogue: 0,0:01:47.95,0:01:50.27,Default,,0000,0000,0000,,A side a side and a side
Dialogue: 0,0:01:50.28,0:01:55.50,Default,,0000,0000,0000,,So we can see that triangle, let me write here with a new color,
Dialogue: 0,0:01:55.51,0:02:09.81,Default,,0000,0000,0000,,ABE is congruent to triangle CBE
Dialogue: 0,0:02:19.62,0:02:21.65,Default,,0000,0000,0000,,Once we know that,
Dialogue: 0,0:02:21.66,0:02:24.55,Default,,0000,0000,0000,,we know that all the corresponding angles are congruent
Dialogue: 0,0:02:24.56,0:02:32.65,Default,,0000,0000,0000,,In particular, we know that angle AEB is going to be congruent
Dialogue: 0,0:02:32.66,0:02:42.37,Default,,0000,0000,0000,,to angle CEB because they are corresponding angles
Dialogue: 0,0:02:42.38,0:02:48.44,Default,,0000,0000,0000,,of congruent triangles
Dialogue: 0,0:02:48.45,0:02:52.05,Default,,0000,0000,0000,,This angle right over here is going to be equal to
Dialogue: 0,0:02:52.06,0:02:53.45,Default,,0000,0000,0000,,that angle over there
Dialogue: 0,0:02:53.46,0:02:55.66,Default,,0000,0000,0000,,We also that they're supplementary
Dialogue: 0,0:03:02.51,0:03:03.34,Default,,0000,0000,0000,,Let me write it this way
Dialogue: 0,0:03:03.35,0:03:06.24,Default,,0000,0000,0000,,They're congruent and they are supplementary
Dialogue: 0,0:03:06.25,0:03:11.35,Default,,0000,0000,0000,,These 2 are gonna have the same measure and
Dialogue: 0,0:03:11.36,0:03:13.28,Default,,0000,0000,0000,,they need to add up to 180 degrees
Dialogue: 0,0:03:13.29,0:03:15.97,Default,,0000,0000,0000,,If I have 2 things that are the same thing and
Dialogue: 0,0:03:15.98,0:03:18.18,Default,,0000,0000,0000,,add up to 180 degrees, what does that tell me?
Dialogue: 0,0:03:18.19,0:03:25.75,Default,,0000,0000,0000,,That tells me that the measure of angle AEB is equal to
Dialogue: 0,0:03:25.76,0:03:31.64,Default,,0000,0000,0000,,the measure of angle CEB which must be equal to 90 degrees
Dialogue: 0,0:03:31.65,0:03:34.73,Default,,0000,0000,0000,,They're the same measure and they're supplementary
Dialogue: 0,0:03:34.74,0:03:37.64,Default,,0000,0000,0000,,This is a right angle and then, this is a right angle
Dialogue: 0,0:03:37.65,0:03:39.36,Default,,0000,0000,0000,,Obviously, this is a right angle
Dialogue: 0,0:03:39.37,0:03:41.32,Default,,0000,0000,0000,,This angle down here is a vertical angle,
Dialogue: 0,0:03:41.33,0:03:42.27,Default,,0000,0000,0000,,that's gonna be a right angle
Dialogue: 0,0:03:42.28,0:03:44.72,Default,,0000,0000,0000,,This is a right angle this over here is gonna be a vertical angle
Dialogue: 0,0:03:44.73,0:03:48.85,Default,,0000,0000,0000,,You see the diagonals intersect at a 90 degree angle
Dialogue: 0,0:03:48.86,0:03:49.94,Default,,0000,0000,0000,,so we've just proved
Dialogue: 0,0:03:49.95,0:03:51.33,Default,,0000,0000,0000,,This is interesting
Dialogue: 0,0:03:51.34,0:03:53.99,Default,,0000,0000,0000,,A parallelogram, the diagonals bisect each other
Dialogue: 0,0:03:54.00,0:03:57.49,Default,,0000,0000,0000,,For a rhombus, where all the sides are equal,
Dialogue: 0,0:03:57.50,0:03:59.87,Default,,0000,0000,0000,,we've shown that not only do they bisect each other
Dialogue: 0,0:03:59.88,0:04:02.81,Default,,0000,0000,0000,,but they're perpendicular bisectors of each other