Congruent triangle example 2

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:02.800
Let's say given this diagram right over here

00:00:02.810 --> 00:00:06.770
We know that the length of segment AB is equal to the length of AC

00:00:06.780 --> 00:00:09.830
so AB which is this whole side right over here

00:00:09.840 --> 00:00:12.880
The length of this entire side as a given

00:00:12.890 --> 00:00:17.950
is equal to the length of this entire side right over here

00:00:18.450 --> 00:00:21.160
So that's the entire side right over there

00:00:21.170 --> 00:00:26.560
And then we also know the angle ABF, ABF is equal to angle ACE

00:00:26.570 --> 00:00:28.680
or you could see their measures are equal or this just implies

00:00:28.690 --> 00:00:31.030
that they are congruent so they have the same measures

00:00:31.040 --> 00:00:36.250
It's equal to angle ACE so this angle right over here is congruent

00:00:36.260 --> 00:00:37.550
to that angle right over there

00:00:37.560 --> 00:00:40.430
Or you could say that they have the same measure

00:00:40.440 --> 00:00:44.070
Now the first thing that I want to attempt to prove in this video

00:00:44.080 --> 00:00:53.570
is whether BF, is whether BF has the same length as CE

00:00:53.580 --> 00:00:57.450
Does BF have the same length as CE

00:00:57.460 --> 00:00:58.450
So let's try to do that

00:00:58.460 --> 00:01:00.080
So we already know a few things I could do

00:01:00.090 --> 00:01:00.970
it two column professionally

00:01:00.980 --> 00:01:02.240
let me just do it just so that

00:01:02.250 --> 00:01:03.690
in case you have to do two column proofs

00:01:03.700 --> 00:01:05.900
in your class you can see how to do it more formally

00:01:05.910 --> 00:01:09.500
So let's try out our statements

00:01:10.230 --> 00:01:15.330
Over here I'm going to write I'm going to write my reason for the statement

00:01:15.340 --> 00:01:18.570
So let me just rewrite this kind of formal two column proof

00:01:18.580 --> 00:01:25.200
so we know AB is equal to AC so this is statement 1 and this is given

00:01:25.500 --> 00:01:33.620
We know statement 2 that angle ABF is equal to angle ACE

00:01:33.630 --> 00:01:36.450
Once again that was given

00:01:36.460 --> 00:01:38.770
Now the other interesting thing on each of these ,we have an angle

00:01:38.780 --> 00:01:43.000
and we have a side, each of these triangles

00:01:43.010 --> 00:01:45.410
And then what you can see is both of the triangles

00:01:45.420 --> 00:01:46.700
and when I say both of the triangles

00:01:46.710 --> 00:01:51.850
I'm talking about triangle ABF and triangle ACE

00:01:51.860 --> 00:01:57.950
and they both share this vertex at A and point A is a vertex for both of these

00:01:57.960 --> 00:02:08.680
So, we could say angle A we say BA, let's call it BAF, angle BAF

00:02:08.690 --> 00:02:10.820
We could say is equal to angle BAF

00:02:10.830 --> 00:02:16.110
or we could say is equal to angle CAE

00:02:16.120 --> 00:02:17.700
That makes it little bit clear that we're dealing

00:02:17.710 --> 00:02:21.190
with two different triangles right here

00:02:21.200 --> 00:02:23.250
But it really is the exact same angle

00:02:23.260 --> 00:02:27.280
It's equal to itself right there that's our third statement

00:02:27.290 --> 00:02:28.620
and we could say that it's obvious

00:02:28.630 --> 00:02:30.840
Some people would call this the reflexive property

00:02:30.850 --> 00:02:33.540
It's obvious that an angle is equal to itself

00:02:33.550 --> 00:02:37.320
And so we could say it's obvious

00:02:37.330 --> 00:02:40.040
or we could call it maybe the reflexive property

00:02:40.050 --> 00:02:44.440
that an angle is clearly reflexive obviously equal to itself

00:02:44.450 --> 00:02:45.710
even if we labeled it different way

00:02:45.720 --> 00:02:48.270
this angle is going to be the same measure

00:02:48.280 --> 00:02:50.410
And now we have something interesting going on,

00:02:50.420 --> 00:02:55.980
we have an angle, a side, and an angle

00:02:55.990 --> 00:03:01.960
So, we end up having is that triangle so by angle-side-angle

00:03:01.970 --> 00:03:07.710
we have the triangle BAF so our statement number 4

00:03:07.720 --> 00:03:10.510
I'm running out of space right here I'll go down right here

00:03:10.520 --> 00:03:17.020
Statement right here is triangle BAF triangle BAF

00:03:17.030 --> 00:03:19.790
Let me kind of highlight it a little more blue right here

00:03:19.800 --> 00:03:25.350
BAF so that's this entire triangle right over here

00:03:25.360 --> 00:03:26.180
And half of the trick of some

00:03:26.190 --> 00:03:28.250
of this problems is seeing the right triangle

00:03:28.260 --> 00:03:30.910
So we started with this white angle we went through the side

00:03:30.920 --> 00:03:33.150
that we knew then we went to this orange angle right over here

00:03:33.160 --> 00:03:36.670
BBA I'm sorry we started at this angle

00:03:36.680 --> 00:03:39.760
then we work to this orange angle across the side E that we know

00:03:39.770 --> 00:03:41.280
is congruent to that side over there

00:03:41.290 --> 00:03:44.140
And then we went to the side the aim of the vertex is not labeled

00:03:44.150 --> 00:03:47.630
So at triangle BAF

00:03:47.640 --> 00:03:52.900
we now know is going to be congruent Congruent to triangle

00:03:53.040 --> 00:03:54.340
we start at the white angle

00:03:54.350 --> 00:03:56.580
go to the orange angle then go to the unlabeled angle

00:03:56.590 --> 00:04:03.470
It's going be congruent to angle to triangle CAF

00:04:03.480 --> 00:04:05.240
So, this is kind of a messily drawn version

00:04:05.240 --> 00:04:07.450
but you can get the idea

00:04:07.460 --> 00:04:09.700
These two triangles are going to be congruent

00:04:09.710 --> 00:04:13.460
C A sorry CAE I should say

00:04:13.470 --> 00:04:17.320
is congruent to triangle CAE

00:04:17.330 --> 00:04:20.730
White angle, orange angle, and then the unlabeled angle

00:04:20.740 --> 00:04:22.920
on that triangle right over there

00:04:22.930 --> 00:04:27.400
And this comes straight out of angle-side-angle

00:04:27.410 --> 00:04:30.860
This comes straight out of ASA

00:04:30.870 --> 00:04:36.090
and these are the two angles and so this is the side in between

00:04:36.100 --> 00:04:40.550
so it comes out of the statements 1, 2, and 3

00:04:40.560 --> 00:04:44.050
And so they're congruent, we know that corresponding sides are going

00:04:44.060 --> 00:04:46.760
to be congruent so we know our statement 5

00:04:46.770 --> 00:04:48.540
We should do this a little bit neater

00:04:48.550 --> 00:04:52.900
Our statement 5, we now know that BF is equal to CE

00:04:53.010 --> 00:04:58.080
BF is equal to CE

00:04:58.090 --> 00:05:00.480
And this comes straight out of statement 4,

00:05:00.490 --> 00:05:05.670
or we could say, corresponding sides,

00:05:05.790 --> 00:05:11.590
sides congruent, corresponding sides are congruent

00:05:11.600 --> 00:05:13.430
Now let's take it up another notch

00:05:13.440 --> 00:05:20.980
Let's see if we can prove whether ED is equal to EF

00:05:20.990 --> 00:05:24.260
So let's just keep going down this and see if we can prove

00:05:24.270 --> 00:05:28.850
whether ED is equal to EF

00:05:28.860 --> 00:05:30.030
I put a question mark there

00:05:30.040 --> 00:05:32.430
'cause we haven't necessarily proven it yet

00:05:32.440 --> 00:05:36.130
So I'm gonna prove that this little short line segment EF is equal to DF

00:05:36.140 --> 00:05:44.830
Sorry not EF is equal to DF, ED is equal to DF

00:05:44.840 --> 00:05:47.670
So let's see if we can prove this right over here

00:05:48.100 --> 00:05:49.460
So the interesting thing that

00:05:49.470 --> 00:05:51.690
we might at first it might not be so obvious

00:05:51.700 --> 00:05:54.920
You know, how do we figure out some type of congruency over that

00:05:54.930 --> 00:05:56.930
but we do already have some information here

00:05:56.940 --> 00:06:00.520
We know that BAF is congruent to CAE

00:06:00.530 --> 00:06:03.950
So we also know that this side right over here

00:06:03.960 --> 00:06:06.280
Let me do it with the color I haven't used yet

00:06:06.290 --> 00:06:10.110
Let me see I have been using a lot of the colors in my pallet

00:06:10.120 --> 00:06:11.400
so it's getting a little too

00:06:11.410 --> 00:06:16.360
So we know from these two congruent triangles that side AE

00:06:16.490 --> 00:06:19.630
Side AE which is part of CAE

00:06:19.640 --> 00:06:26.870
We know that AE is going to be equal to AF

00:06:26.880 --> 00:06:29.680
That these two sides are congruent and the reason why is

00:06:29.690 --> 00:06:32.780
'cause they're corresponding sides of congruent triangles

00:06:32.790 --> 00:06:38.320
AF is the side opposite the white angle on BAF triangle BAF

00:06:38.330 --> 00:06:43.180
And AE is the side opposite the white angle on triangle CAE

00:06:43.190 --> 00:06:44.340
which we know are congruent

00:06:44.350 --> 00:06:49.990
So we know that AE is equal to AF

00:06:50.000 --> 00:06:52.750
And once again this comes from statement 4

00:06:52.760 --> 00:06:55.570
and we could even say corresponding sides congruent

00:06:55.580 --> 00:06:58.250
Same reason as we gave right up here

00:06:58.270 --> 00:07:00.770
Now, what's interesting here is you know

00:07:00.780 --> 00:07:02.730
this isn't even a triangle that we're seeing up here

00:07:02.740 --> 00:07:04.730
but this information that these two characters

00:07:04.740 --> 00:07:08.090
are congruent help us with this part over here

00:07:08.100 --> 00:07:18.850
Because we know that BA or as to say we know that AB is equal to AC

00:07:18.860 --> 00:07:23.560
that was given and so we know that EB

00:07:23.570 --> 00:07:24.910
Let me write it over here

00:07:24.920 --> 00:07:27.400
and make it a little bit messy right over here

00:07:27.410 --> 00:07:31.020
Statement 7 will give us will give us some space

00:07:31.030 --> 00:07:36.280
we know that BE is going to be equal to CF

00:07:36.280 --> 00:07:41.600
Let me write that down, we know that BE is equal to CF

00:07:41.610 --> 00:07:45.200
And why do we know that let me put the reason right over here

00:07:45.210 --> 00:07:47.970
Let me try to clean up my work a little bit

00:07:47.980 --> 00:07:50.890
This column has been slowly drifting to the left

00:07:50.900 --> 00:07:54.260
But how do we know that BE is equal to CF?

00:07:54.270 --> 00:08:09.600
Well we know that the length of BE is equal to the length of BA minus AE

00:08:09.610 --> 00:08:11.480
or I could just say AB I could that's how

00:08:11.490 --> 00:08:13.560
I call it up here so it's equal to

00:08:13.940 --> 00:08:22.670
AB minus AE is the same thing

00:08:22.680 --> 00:08:25.460
based on these last few things that we saw

00:08:25.470 --> 00:08:31.950
As saying AC minus AF cause AB is equal to AC so that's equal to AC

00:08:31.960 --> 00:08:34.760
and AE we already showed as the same thing as AF

00:08:34.770 --> 00:08:43.480
AC minus AF and AC minus AF is the same thing as CF right over here

00:08:43.490 --> 00:08:46.610
is equal to CF right over there

00:08:46.620 --> 00:08:52.230
And we know that because and we know this from statement 1,

00:08:52.240 --> 00:08:58.420
we know it from statement 5 and we know it from statement 6

00:08:58.430 --> 00:09:00.740
Actually we didn't need we didn't need statement 5 there

00:09:00.750 --> 00:09:02.770
Let me see we just need 1 and 6

00:09:02.780 --> 00:09:07.180
So, let's say we need this is from 1 and 6 is what we had to do there

00:09:07.190 --> 00:09:10.300
So, we just know that look this side is equal to that side

00:09:10.310 --> 00:09:11.810
This little part is equal to that part

00:09:11.820 --> 00:09:14.070
So if you subtract the big part minus the little part

00:09:14.080 --> 00:09:16.920
This right over here is going to be equal to this right over here

00:09:16.930 --> 00:09:17.820
So, that's all we're showing

00:09:17.830 --> 00:09:23.000
So, this yellow side is equal to this yellow side right over here

00:09:23.010 --> 00:09:24.530
Now the other thing that we know

00:09:24.540 --> 00:09:28.930
And this is straight out of vertical angles is that this angle EDB

00:09:28.940 --> 00:09:31.880
is going to be congruent to angle FDC

00:09:31.890 --> 00:09:33.220
So let me write that down again

00:09:33.230 --> 00:09:48.610
So 8, we know that angle EDB is going to be equal to angle FDC

00:09:48.620 --> 00:09:55.230
That comes straight out of vertical angles are congruent

00:09:55.240 --> 00:09:56.780
or their measures are equal

00:09:56.790 --> 00:09:59.590
And now all of a sudden we have something interesting again

00:09:59.600 --> 00:10:04.690
We have the Orange angle-white angle-side

00:10:04.700 --> 00:10:08.340
So we know that these two smaller triangles are congruent

00:10:08.350 --> 00:10:11.080
So, now we know I don't want to lose my diagram

00:10:11.090 --> 00:10:15.490
We know that triangle BED, we know so statement number 9

00:10:15.500 --> 00:10:20.660
we know that triangle BED is congruent

00:10:20.670 --> 00:10:27.260
So, BED is this one we know that BED is congruent to triangle

00:10:27.270 --> 00:10:30.570
Now we want to use the same sides white angle yellow side

00:10:30.580 --> 00:10:31.550
then orange angle

00:10:31.560 --> 00:10:37.580
White angle white angle let me be careful here, white angle

00:10:38.560 --> 00:10:41.880
So B is white angle, E is the unlabeled angle,

00:10:41.890 --> 00:10:44.770
and then D is the labeled angle the orange labeled angle

00:10:44.780 --> 00:10:48.010
So, you want to start C unlabeled angle orange angle

00:10:48.020 --> 00:10:53.400
so triangle CFD

00:10:53.410 --> 00:10:55.580
And this comes straight from once again,

00:10:55.590 --> 00:10:57.690
orange angle white angle side

00:10:57.700 --> 00:11:00.040
So angle-angle-side orange sorry,

00:11:00.040 --> 00:11:02.210
orange angle white angle side

00:11:02.220 --> 00:11:07.680
So this come straight angle-angle-side congruency

00:11:07.690 --> 00:11:09.750
And since we've now shown that this triangle is equal

00:11:09.760 --> 00:11:13.100
to that triangle we know that their corresponding sides are equal

00:11:13.110 --> 00:11:14.640
And then this is our homestretch

00:11:14.650 --> 00:11:18.260
We now know since these two triangles are congruent

00:11:18.270 --> 00:11:23.000
We now know that ED is equal to DF

00:11:23.010 --> 00:11:24.390
because they're corresponding sides

00:11:24.400 --> 00:11:28.560
And I could write that right over here ED is equal to DF

00:11:28.570 --> 00:11:31.070
And once again the reason here is

00:11:31.080 --> 00:11:33.650
the same thing up here corresponding

00:11:33.660 --> 00:11:38.130
so we know our statement 9 which means they're congruent

00:11:38.140 --> 00:11:44.650
And corresponding sides congruent and we are done

00:11:44.660 --> 00:11:46.240
So that was a pretty involved problem

00:11:46.250 --> 00:11:47.860
But you see once again you go step by step

00:11:47.870 --> 00:11:51.290
just try to figure out each triangle and you eventually get it

00:11:51.300 --> 00:11:54.710
But really the hard part isn't so much the realizing which postulate

00:11:54.720 --> 00:11:57.410
to use or how to apply them necessarily

00:11:57.420 --> 00:11:59.060
But seeing the triangle seeing that

00:11:59.060 --> 00:12:00.390
there's some information there

00:12:00.400 --> 00:12:06.900
Seeing that you could figure out BE by subtracting it from AE minus BE

00:12:06.910 --> 00:12:10.350
Seeing that there are two triangles kind of overlapping in the star

00:12:10.360 --> 00:12:13.250
or arms or whatever you might want to call it


Congruent triangle proof example

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:04.690
So we have these two parallel
lines, line segment AB

00:00:04.690 --> 00:00:07.560
and line segment CD.

00:00:07.560 --> 00:00:09.810
I should say they are
parallel line segments.

00:00:09.810 --> 00:00:11.460
And then we have
these transversals

00:00:11.460 --> 00:00:12.660
that go across them.

00:00:12.660 --> 00:00:15.020
So you have this transversal
BC right over here.

00:00:15.020 --> 00:00:17.020
And you have this
transversal AD.

00:00:17.020 --> 00:00:18.540
And what this
diagram tells us is

00:00:18.540 --> 00:00:21.610
that the distance between A
and E-- this little hash mark--

00:00:21.610 --> 00:00:24.490
says that this line segment
is the same distance

00:00:24.490 --> 00:00:26.610
as the distance between
E and D. Or another way

00:00:26.610 --> 00:00:30.640
to think about it is that
point E is at the midpoint,

00:00:30.640 --> 00:00:34.640
or is the midpoint,
of line segment AD.

00:00:34.640 --> 00:00:37.320
And what I want to think
about in this video

00:00:37.320 --> 00:00:43.430
is, is point E also the
midpoint of line segment BC?

00:00:43.430 --> 00:00:46.530
So this is the question
right over here.

00:00:46.530 --> 00:00:58.576
So is E the midpoint
of line segment BC?

00:00:58.576 --> 00:01:00.950
And you could imagine, based
on a lot of the videos we've

00:01:00.950 --> 00:01:02.830
been seeing lately,
maybe it has something

00:01:02.830 --> 00:01:05.239
to do with congruent triangles.

00:01:05.239 --> 00:01:08.130
So let's see if we can set up
some congruency relationship

00:01:08.130 --> 00:01:10.420
between the two obvious
triangles in this diagram.

00:01:10.420 --> 00:01:12.300
We have this triangle
up here on the left.

00:01:12.300 --> 00:01:13.770
And then we have this
triangle down here.

00:01:13.770 --> 00:01:15.644
This one kind of looks
like it's pointing up.

00:01:15.644 --> 00:01:18.040
This one looks like
it's pointing down.

00:01:18.040 --> 00:01:20.940
So there's a bunch of things
we know about vertical angles

00:01:20.940 --> 00:01:23.380
and angles of transversals.

00:01:23.380 --> 00:01:26.390
The most obvious one is
that we have this vertical.

00:01:26.390 --> 00:01:30.292
We know that angle AEB
is going to be congruent.

00:01:30.292 --> 00:01:32.750
Or its measure is going to be
equal to the measure of angle

00:01:32.750 --> 00:01:34.370
CED.

00:01:34.370 --> 00:01:45.250
So we know that
angle AEB is going

00:01:45.250 --> 00:01:51.700
to be congruent to angle
DEC, which really just means

00:01:51.700 --> 00:01:54.020
they have the
exact same measure.

00:01:54.020 --> 00:01:57.245
And we know that because
they are vertical angles.

00:02:03.530 --> 00:02:06.850
Now, we also know that
AB and CD are parallel.

00:02:06.850 --> 00:02:11.115
So this line right over
here, this is a transversal.

00:02:11.115 --> 00:02:12.490
And there's actually
several ways

00:02:12.490 --> 00:02:14.370
that we can do this problem.

00:02:14.370 --> 00:02:17.035
But we know that this
is a transversal.

00:02:17.035 --> 00:02:19.660
And there's a couple of ways to
think about it right over here.

00:02:19.660 --> 00:02:21.130
So let me just continue
the transversal,

00:02:21.130 --> 00:02:23.004
so we get to see all of
the different angles.

00:02:23.004 --> 00:02:27.910
You could say that this
angle right here, angle

00:02:27.910 --> 00:02:33.560
ABE-- so this is its measure
right over here-- you could say

00:02:33.560 --> 00:02:37.550
that it is the alternate
interior angle to angle ECD,

00:02:37.550 --> 00:02:39.510
to this angle right over there.

00:02:39.510 --> 00:02:41.321
And if that didn't
jump out at you,

00:02:41.321 --> 00:02:43.570
you would say that the
corresponding angle to this one

00:02:43.570 --> 00:02:46.340
right over here is this
angle right up here.

00:02:46.340 --> 00:02:48.822
If you were to continue
this line off a little bit,

00:02:48.822 --> 00:02:50.280
these are the
corresponding angles.

00:02:50.280 --> 00:02:51.530
And then this one is vertical.

00:02:51.530 --> 00:02:58.950
But either way, angle
ABE-- let me be careful.

00:02:58.950 --> 00:03:07.633
Angle ABE is going to be
congruent to angle DCE.

00:03:12.210 --> 00:03:14.840
And we could say because it's
alternate interior angles.

00:03:17.712 --> 00:03:19.170
I'll just write a
little code here.

00:03:19.170 --> 00:03:27.500
So Alt interior angles.

00:03:27.500 --> 00:03:29.610
And then we have an
interesting relationship.

00:03:29.610 --> 00:03:32.860
We have an angle congruent to an
angle, another angle congruent

00:03:32.860 --> 00:03:33.360
to an angle.

00:03:33.360 --> 00:03:36.200
And then the next side is
congruent to the next side

00:03:36.200 --> 00:03:36.700
over here.

00:03:36.700 --> 00:03:38.660
So pink, green, side.

00:03:38.660 --> 00:03:40.700
Pink, green, side.

00:03:40.700 --> 00:03:44.700
So we can employ AAS,
angle-angle-side.

00:03:44.700 --> 00:03:46.180
And it's in the right order.

00:03:46.180 --> 00:03:47.597
So now, we know
that triangle-- we

00:03:47.597 --> 00:03:50.138
have to make sure that we get
the letters right here, that we

00:03:50.138 --> 00:03:51.910
have the right
corresponding vertices.

00:03:51.910 --> 00:03:58.060
We can say that
triangle AEB-- actually,

00:03:58.060 --> 00:04:00.680
let me start with the angle
just to make it interesting.

00:04:00.680 --> 00:04:03.450
Angle BEA, so we're starting
with the magenta angle,

00:04:03.450 --> 00:04:04.880
going to the green angle,
and then going to the one

00:04:04.880 --> 00:04:05.970
that we haven't labeled.

00:04:05.970 --> 00:04:11.890
So angle BEA, we can say,
is congruent to angle--

00:04:11.890 --> 00:04:14.950
we start with the
magenta vertices-- C,

00:04:14.950 --> 00:04:19.920
go to the center, E, and then
go the unlabeled one, D. And we

00:04:19.920 --> 00:04:21.910
know this because
of angle-angle-side.

00:04:21.910 --> 00:04:25.260
And they correspond to each
other-- magenta-green-side,

00:04:25.260 --> 00:04:26.320
magenta-green-side.

00:04:26.320 --> 00:04:27.560
They're all congruent.

00:04:27.560 --> 00:04:31.520
So this is from AAS.

00:04:31.520 --> 00:04:33.890
And then, if we know
that they are congruent,

00:04:33.890 --> 00:04:36.480
then that means corresponding
sides are congruent.

00:04:36.480 --> 00:04:42.357
So then we know these two
triangles are congruent.

00:04:42.357 --> 00:04:44.190
So that means that their
corresponding sides

00:04:44.190 --> 00:04:45.220
are congruent.

00:04:45.220 --> 00:04:52.689
So then we know that length
of BE is going to be equal--

00:04:52.689 --> 00:04:54.730
and that's the segment
that's between the magenta

00:04:54.730 --> 00:04:56.530
and the green angles.

00:04:56.530 --> 00:05:00.060
The corresponding side is
side CE between the magenta

00:05:00.060 --> 00:05:04.037
and the green angles--
is equal to CE.

00:05:04.037 --> 00:05:06.120
And this just comes out
of the previous statement.

00:05:06.120 --> 00:05:09.620
If we number them, that's
1, that's 2, and that's 3.

00:05:09.620 --> 00:05:12.600
And so that comes
out of statement 3.

00:05:12.600 --> 00:05:13.980
And so we have proven this.

00:05:13.980 --> 00:05:15.520
E is the midpoint of BC.

00:05:15.520 --> 00:05:19.120
It comes straight out of the
fact that BE is equal to CE.

00:05:19.120 --> 00:05:20.710
So I can mark this
off with hash.

00:05:20.710 --> 00:05:22.687
This line segment
right over here

00:05:22.687 --> 00:05:24.770
is congruent to this line
segment right over here,

00:05:24.770 --> 00:05:28.002
because we know that those
two triangles are congruent.

00:05:28.002 --> 00:05:29.460
And I've inadvertently,
right here,

00:05:29.460 --> 00:05:31.180
done a little two-column proof.

00:05:31.180 --> 00:05:33.735
This over here on the
left-hand side is my statement.

00:05:37.700 --> 00:05:43.840
And then on the right-hand
side, I gave my reason.

00:05:43.840 --> 00:05:45.800
And we're done.


More on why SSA is not a postulate

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.170
Several videos
ago, I very quickly

00:00:02.170 --> 00:00:08.322
went through why side-side-angle
is not a valid postulate.

00:00:08.322 --> 00:00:09.780
And what I want to
do in this video

00:00:09.780 --> 00:00:11.450
is explore it a little bit more.

00:00:11.450 --> 00:00:13.380
And it's not called
angle-side-side

00:00:13.380 --> 00:00:15.530
for obvious reasons,
because then the acronym

00:00:15.530 --> 00:00:18.352
would make people giggle
in geometry class.

00:00:18.352 --> 00:00:20.060
And I guess we don't
want people giggling

00:00:20.060 --> 00:00:21.870
while they're doing mathematics.

00:00:21.870 --> 00:00:24.720
So let's just think
about a triangle here.

00:00:24.720 --> 00:00:26.300
So let's say I have a triangle.

00:00:26.300 --> 00:00:27.940
Let me draw it.

00:00:27.940 --> 00:00:31.065
Let's have a triangle that
looks something like this.

00:00:31.065 --> 00:00:33.525
If I have a triangle
that looks something--

00:00:33.525 --> 00:00:35.340
I have trouble drawing
straight triangles.

00:00:35.340 --> 00:00:38.220
So let's say the triangle
looks something like that.

00:00:41.210 --> 00:00:45.570
And let's say that we've
found another triangle that

00:00:45.570 --> 00:00:48.000
has a congruent
side, a side that

00:00:48.000 --> 00:00:49.970
is congruent to this
side right over here.

00:00:53.770 --> 00:00:56.710
I guess any side on a triangle
is next to the other two sides.

00:00:56.710 --> 00:00:59.340
Next to that is a side that
is congruent to this side

00:00:59.340 --> 00:01:00.830
right over here.

00:01:00.830 --> 00:01:05.310
And then that side is one
of the sides of an angle.

00:01:05.310 --> 00:01:08.060
So it forms one of the parts
of an angle right over here.

00:01:08.060 --> 00:01:10.700
And that other triangle
has a congruent angle right

00:01:10.700 --> 00:01:11.200
over here.

00:01:11.200 --> 00:01:15.740
So this is the angle that that
first side is not a part of.

00:01:15.740 --> 00:01:17.860
Only that second side
is part of this angle.

00:01:17.860 --> 00:01:19.640
So this is side-side-angle.

00:01:19.640 --> 00:01:22.180
Or you could call
it angle-side-side

00:01:22.180 --> 00:01:24.140
and giggle a little
bit about it.

00:01:24.140 --> 00:01:27.280
Now, how do we know that
this doesn't by itself

00:01:27.280 --> 00:01:28.447
show that this is congruent?

00:01:28.447 --> 00:01:30.446
Well, we'd have to show
that this could actually

00:01:30.446 --> 00:01:31.880
imply two different triangles.

00:01:31.880 --> 00:01:33.930
And to think about
that, let's say

00:01:33.930 --> 00:01:36.550
we know that the angle-- we know
that this other triangle has

00:01:36.550 --> 00:01:39.090
that same yellow
angle there, which

00:01:39.090 --> 00:01:42.800
means that the blue side is
going to have to look something

00:01:42.800 --> 00:01:48.020
like that, just the way
we drew it over here.

00:01:48.020 --> 00:01:50.500
This side down here, I'll
make it a green side.

00:01:50.500 --> 00:01:53.400
This green side down here
we know nothing about.

00:01:53.400 --> 00:01:55.576
We never said that this
is congruent to anything.

00:01:55.576 --> 00:01:57.450
If we knew, then we
could use side-side-side.

00:01:57.450 --> 00:01:58.670
We only know that
this side is congruent

00:01:58.670 --> 00:02:01.160
and this side is congruent,
and this angle is congruent.

00:02:01.160 --> 00:02:03.460
So this green side, and I'll
draw it as a dotted line,

00:02:03.460 --> 00:02:04.543
it could be of any length.

00:02:04.543 --> 00:02:08.570
We don't know what the
length is of that green side.

00:02:08.570 --> 00:02:10.880
Now we have this magenta side.

00:02:10.880 --> 00:02:13.171
We have another side
that is congruent here.

00:02:13.171 --> 00:02:14.670
So this thing could
pivot over here.

00:02:14.670 --> 00:02:17.670
We know nothing about this angle
so it could form any angle.

00:02:17.670 --> 00:02:19.720
But it does have to
get to this other side.

00:02:19.720 --> 00:02:22.440
So one possibility is that maybe
the triangles are congruent.

00:02:22.440 --> 00:02:24.310
So maybe this side
does go down just

00:02:24.310 --> 00:02:26.360
like that, in which
case, we actually

00:02:26.360 --> 00:02:28.160
would have congruent triangles.

00:02:28.160 --> 00:02:30.340
But the kind of aha
moment here, or the reason

00:02:30.340 --> 00:02:33.290
why SSA isn't possible, is
that this side, could also

00:02:33.290 --> 00:02:34.295
come down like this.

00:02:37.340 --> 00:02:39.910
There's two ways to get
down to this base, if you

00:02:39.910 --> 00:02:41.460
want to call it that way.

00:02:41.460 --> 00:02:44.870
It can come out that way or it
could kind of come in this way.

00:02:44.870 --> 00:02:49.660
And so that's why SSA by itself
with no other information

00:02:49.660 --> 00:02:50.860
is ambiguous.

00:02:50.860 --> 00:02:54.590
It does not give you
enough information

00:02:54.590 --> 00:02:58.010
to say that those triangles
are definitely the same.

00:02:58.010 --> 00:02:59.710
Now there are special cases.

00:02:59.710 --> 00:03:02.390
So in this situation
right over here,

00:03:02.390 --> 00:03:07.240
our angle, the angle in our
SSA, our angle was acute.

00:03:07.240 --> 00:03:11.185
This is an acute
angle right over here.

00:03:11.185 --> 00:03:12.810
And when you have an
acute angle as one

00:03:12.810 --> 00:03:14.920
of the sides of your
triangle, the other sides

00:03:14.920 --> 00:03:17.350
of the triangle, you could
still have an obtuse angle.

00:03:17.350 --> 00:03:18.830
Remember, acute
means less than 90

00:03:18.830 --> 00:03:21.770
degrees, obtuse means
greater than 90 degrees.

00:03:21.770 --> 00:03:23.850
So you could still
have an obtuse angle.

00:03:23.850 --> 00:03:25.750
So that's why this is an option.

00:03:25.750 --> 00:03:28.380
So one option is that you
have two other acute angles.

00:03:28.380 --> 00:03:30.270
So this could be acute.

00:03:30.270 --> 00:03:32.414
This is also acute,
also acute, also acute.

00:03:32.414 --> 00:03:33.830
But then you have
the option where

00:03:33.830 --> 00:03:35.820
this is even more
acute, even narrower,

00:03:35.820 --> 00:03:40.010
and then this becomes
an obtuse angle.

00:03:40.010 --> 00:03:42.700
And that's only possible if
you don't-- you can't have two

00:03:42.700 --> 00:03:44.300
obtuse angles in
the same triangle.

00:03:44.300 --> 00:03:46.050
You can't have two
things that have larger

00:03:46.050 --> 00:03:49.430
than 90-degree measure
in the same triangle.

00:03:49.430 --> 00:03:52.000
And so that's why
there is a possibility

00:03:52.000 --> 00:03:59.120
where if you have another
triangle that looks like this,

00:03:59.120 --> 00:04:02.650
and if I were to tell you
very clearly that this angle

00:04:02.650 --> 00:04:10.190
right over here is obtuse--
and that is the A in the SSA.

00:04:10.190 --> 00:04:11.650
So you have the angle.

00:04:11.650 --> 00:04:15.220
And I were to say I have another
triangle where this angle is

00:04:15.220 --> 00:04:17.329
congruent to that other
triangle, some angle

00:04:17.329 --> 00:04:20.190
of that other triangle, and then
one of the sides adjacent to it

00:04:20.190 --> 00:04:25.690
is congruent, and then the next
side over is also congruent,

00:04:25.690 --> 00:04:27.000
then it's not so ambiguous.

00:04:27.000 --> 00:04:28.830
Because we could
try to draw that.

00:04:28.830 --> 00:04:33.050
So let's draw that same
congruent obtuse angle.

00:04:33.050 --> 00:04:35.274
We know nothing about
this side down here

00:04:35.274 --> 00:04:37.690
because we haven't said that
that's necessarily congruent.

00:04:37.690 --> 00:04:41.120
So that could be of any length.

00:04:41.120 --> 00:04:43.650
We do know that this
triangle is going

00:04:43.650 --> 00:04:48.430
to have the same length
for this side of the angle.

00:04:48.430 --> 00:04:51.830
So it looks like this.

00:04:51.830 --> 00:04:54.020
And then we know
that this side--

00:04:54.020 --> 00:04:56.470
let me do that in orange.

00:04:56.470 --> 00:04:59.995
We know that this side is also
going to be the same length.

00:04:59.995 --> 00:05:01.870
We haven't told you
anything about this angle

00:05:01.870 --> 00:05:02.950
right over here.

00:05:02.950 --> 00:05:05.480
So this side could
pivot over here.

00:05:05.480 --> 00:05:07.220
We can kind of
rotate it over there.

00:05:07.220 --> 00:05:09.930
But there's only one way,
now, that this orange side

00:05:09.930 --> 00:05:11.430
can reach this green side.

00:05:11.430 --> 00:05:14.650
Now the only way is
this way over here.

00:05:14.650 --> 00:05:17.280
And we were more constrained,
or this case isn't ambiguous,

00:05:17.280 --> 00:05:19.320
because we used up
our obtuse angle here.

00:05:19.320 --> 00:05:21.480
The A here is an obtuse one.

00:05:21.480 --> 00:05:25.820
And so then it constrains
what the triangle can become.

00:05:25.820 --> 00:05:30.050
So I don't want to make
you say, in general, SSA,

00:05:30.050 --> 00:05:33.012
you do not want to
use it as a postulate.

00:05:33.012 --> 00:05:34.720
I just wanted to make
it clear that there

00:05:34.720 --> 00:05:37.460
is the special case where if
you know that the A in the SSA

00:05:37.460 --> 00:05:41.224
is obtuse, then it becomes
a little bit less ambiguous.

00:05:41.224 --> 00:05:42.890
And then finally,
there's a circumstance

00:05:42.890 --> 00:05:45.775
that this is an acute angle
where it would be ambiguous.

00:05:45.775 --> 00:05:47.400
You have the obtuse
angle, and then you

00:05:47.400 --> 00:05:49.740
have something in between,
which is the right angle.

00:05:49.740 --> 00:05:53.600
So where you have the A
in SSA is a right angle.

00:05:53.600 --> 00:05:55.440
So if you had it like this.

00:05:55.440 --> 00:05:59.040
If you have a
right angle and you

00:05:59.040 --> 00:06:04.580
have some base of unknown length
but you fix this length right

00:06:04.580 --> 00:06:07.750
over here-- if you know that
this is fixed because you're

00:06:07.750 --> 00:06:09.830
saying it's congruent
to some other triangle,

00:06:09.830 --> 00:06:12.380
and if you know that the
next length is fixed--

00:06:12.380 --> 00:06:14.099
and if you think about
it, this next side

00:06:14.099 --> 00:06:16.140
is going to be the side
opposite the right angle.

00:06:16.140 --> 00:06:18.270
It's going to have to be the
hypotenuse of the right angle.

00:06:18.270 --> 00:06:20.860
Then you know that the only
way you can construct this,

00:06:20.860 --> 00:06:23.940
and similar to the
obtuse case, and if you

00:06:23.940 --> 00:06:26.280
know the length of this,
the only way you could do it

00:06:26.280 --> 00:06:28.170
is to bring it down over here.

00:06:28.170 --> 00:06:30.170
So that actually does
lead to another postulate

00:06:30.170 --> 00:06:36.810
called the right angle side
hypotenuse postulate, which

00:06:36.810 --> 00:06:40.090
is really just a
special case of SSA

00:06:40.090 --> 00:06:44.599
where the angle is
actually a right angle.

00:06:44.599 --> 00:06:46.140
And here, they wrote
the angle first.

00:06:46.140 --> 00:06:48.097
You could view this
as angle-side-side.

00:06:48.097 --> 00:06:49.930
And they were able to
do it because now they

00:06:49.930 --> 00:06:51.679
can write "right angle,"
and so it doesn't

00:06:51.679 --> 00:06:54.699
form that embarrassing acronym.

00:06:54.699 --> 00:06:56.740
And this would also be a
little bit common sense.

00:06:56.740 --> 00:06:59.390
Because if you know two
sides of a right triangle--

00:06:59.390 --> 00:07:01.640
and we haven't gone into
depth on this in the geometry

00:07:01.640 --> 00:07:04.040
playlist, but you might
already be familiar with it--

00:07:04.040 --> 00:07:06.160
by Pythagorean
theorem, you can always

00:07:06.160 --> 00:07:07.690
figure out the third side.

00:07:07.690 --> 00:07:10.220
So if you have this
information about any triangle,

00:07:10.220 --> 00:07:12.610
you can always figure
out the third side.

00:07:12.610 --> 00:07:14.420
And then you can
use side-side-side.

00:07:14.420 --> 00:07:16.670
So I just wanted to show you
this little special case.

00:07:16.670 --> 00:07:18.128
But in general,
the important thing

00:07:18.128 --> 00:07:21.380
is that you can't just
use SSA unless you

00:07:21.380 --> 00:07:23.394
have more information.


Finding congruent triangles

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.700
What we have drawn over here
is five different triangles.

00:00:04.700 --> 00:00:06.530
And what I want to
do in this video

00:00:06.530 --> 00:00:08.660
is figure out which
of these triangles

00:00:08.660 --> 00:00:11.360
are congruent to which
other of these triangles.

00:00:11.360 --> 00:00:13.110
And to figure that
out, I'm just over here

00:00:13.110 --> 00:00:16.466
going to write our triangle
congruency postulate.

00:00:16.466 --> 00:00:17.840
So we know that
two triangles are

00:00:17.840 --> 00:00:20.300
congruent if all of their
sides are the same-- so

00:00:20.300 --> 00:00:21.940
side, side, side.

00:00:21.940 --> 00:00:25.170
We also know they are congruent
if we have a side and then

00:00:25.170 --> 00:00:27.960
an angle between the sides
and then another side that

00:00:27.960 --> 00:00:30.170
is congruent-- so
side, angle, side.

00:00:30.170 --> 00:00:31.980
If we reverse the
angles and the sides,

00:00:31.980 --> 00:00:34.110
we know that's also a
congruence postulate.

00:00:34.110 --> 00:00:36.650
So if we have an angle
and then another angle

00:00:36.650 --> 00:00:40.080
and then the side in
between them is congruent,

00:00:40.080 --> 00:00:42.360
then we also have two
congruent triangles.

00:00:42.360 --> 00:00:44.280
And then finally, if we
have an angle and then

00:00:44.280 --> 00:00:47.000
another angle and
then a side, then that

00:00:47.000 --> 00:00:49.790
is also-- any of these
imply congruency.

00:00:49.790 --> 00:00:51.565
So let's see our
congruent triangles.

00:00:51.565 --> 00:00:53.690
So let's see what we can
figure out right over here

00:00:53.690 --> 00:00:55.350
for these triangles.

00:00:55.350 --> 00:00:58.710
So right in this
triangle ABC over here,

00:00:58.710 --> 00:01:01.910
we're given this length 7,
then 60 degrees, and then

00:01:01.910 --> 00:01:02.460
40 degrees.

00:01:02.460 --> 00:01:03.834
Or another way to
think about it,

00:01:03.834 --> 00:01:07.480
we're given an angle, an angle
and a side-- 40 degrees, then

00:01:07.480 --> 00:01:09.760
60 degrees, then 7.

00:01:09.760 --> 00:01:11.950
And in order for something
to be congruent here,

00:01:11.950 --> 00:01:14.300
they would have to have an
angle, angle, side given--

00:01:14.300 --> 00:01:16.200
at least, unless maybe
we have to figure it

00:01:16.200 --> 00:01:17.270
out some other way.

00:01:17.270 --> 00:01:18.790
But I'm guessing
for this problem,

00:01:18.790 --> 00:01:20.590
they'll just already
give us the angle.

00:01:20.590 --> 00:01:22.190
So they'll have to have an
angle, an angle, and side.

00:01:22.190 --> 00:01:24.220
And it can't just be any
angle, angle, and side.

00:01:24.220 --> 00:01:28.260
It has to be 40, 60, and 7, and
it has to be in the same order.

00:01:28.260 --> 00:01:30.850
It can't be 60 and
then 40 and then 7.

00:01:30.850 --> 00:01:33.710
If the 40-degree side
has-- if one of its sides

00:01:33.710 --> 00:01:36.610
has the length 7, then that
is not the same thing here.

00:01:36.610 --> 00:01:39.777
Here, the 60-degree
side has length 7.

00:01:39.777 --> 00:01:41.610
So let's see if any of
these other triangles

00:01:41.610 --> 00:01:44.220
have this kind of 40,
60 degrees, and then

00:01:44.220 --> 00:01:46.190
the 7 right over here.

00:01:46.190 --> 00:01:48.730
So this has the 40 degrees
and the 60 degrees,

00:01:48.730 --> 00:01:49.939
but the 7 is in between them.

00:01:49.939 --> 00:01:51.604
So this looks like
it might be congruent

00:01:51.604 --> 00:01:53.780
to some other triangle,
maybe closer to something

00:01:53.780 --> 00:01:55.280
like angle, side,
angle because they

00:01:55.280 --> 00:01:56.550
have an angle, side, angle.

00:01:56.550 --> 00:01:58.060
So it wouldn't be that one.

00:01:58.060 --> 00:01:59.320
This one looks interesting.

00:01:59.320 --> 00:02:00.842
This is also angle, side, angle.

00:02:00.842 --> 00:02:03.300
So maybe these are congruent,
but we'll check back on that.

00:02:03.300 --> 00:02:05.870
We're still focused on
this one right over here.

00:02:05.870 --> 00:02:09.190
And this one, we have a 60
degrees, then a 40 degrees,

00:02:09.190 --> 00:02:10.340
and a 7.

00:02:10.340 --> 00:02:11.110
This is tempting.

00:02:11.110 --> 00:02:14.074
We have an angle, an
angle, and a side,

00:02:14.074 --> 00:02:15.740
but the angles are
in a different order.

00:02:15.740 --> 00:02:17.450
Here it's 40, 60, 7.

00:02:17.450 --> 00:02:19.629
Here it's 60, 40, 7.

00:02:19.629 --> 00:02:21.170
So it's an angle,
an angle, and side,

00:02:21.170 --> 00:02:25.460
but the side is not on
the 60-degree angle.

00:02:25.460 --> 00:02:27.270
It's on the 40-degree
angle over here.

00:02:27.270 --> 00:02:29.040
So this doesn't
look right either.

00:02:29.040 --> 00:02:32.880
Here we have 40 degrees,
60 degrees, and then 7.

00:02:32.880 --> 00:02:34.870
So this is looking pretty good.

00:02:34.870 --> 00:02:37.670
We have this side
right over here

00:02:37.670 --> 00:02:41.310
is congruent to this
side right over here.

00:02:41.310 --> 00:02:47.754
Then you have your 60-degree
angle right over here.

00:02:47.754 --> 00:02:49.670
It might not be obvious,
because it's flipped,

00:02:49.670 --> 00:02:51.125
and they're drawn a
little bit different.

00:02:51.125 --> 00:02:53.170
But you should never assume
that just the drawing

00:02:53.170 --> 00:02:54.770
tells you what's going on.

00:02:54.770 --> 00:02:57.860
And then finally, you have
your 40-degree angle here,

00:02:57.860 --> 00:03:00.220
which is your
40-degree angle here.

00:03:00.220 --> 00:03:03.924
So we can say-- we can
write down-- and let

00:03:03.924 --> 00:03:05.340
me think of a good
place to do it.

00:03:05.340 --> 00:03:06.830
I'll write it right over here.

00:03:06.830 --> 00:03:13.345
We can write down that triangle
ABC is congruent to triangle--

00:03:13.345 --> 00:03:16.040
and now we have to be very
careful with how we name this.

00:03:16.040 --> 00:03:17.415
We have to make
sure that we have

00:03:17.415 --> 00:03:21.280
the corresponding
vertices map up together.

00:03:21.280 --> 00:03:25.210
So for example, we started
this triangle at vertex A.

00:03:25.210 --> 00:03:27.030
So point A right
over here, that's

00:03:27.030 --> 00:03:28.660
where we have the
60-degree angle.

00:03:28.660 --> 00:03:31.030
That's the vertex of
the 60-degree angle.

00:03:31.030 --> 00:03:34.490
So the vertex of the 60-degree
angle over here is point N.

00:03:34.490 --> 00:03:36.490
So I'm going to go to N.

00:03:36.490 --> 00:03:40.630
And then we went from A to B. B
was the vertex that we did not

00:03:40.630 --> 00:03:41.990
have any angle for.

00:03:41.990 --> 00:03:43.170
And we could figure it out.

00:03:43.170 --> 00:03:44.919
If these two guys add
up to 100, then this

00:03:44.919 --> 00:03:46.840
is going to be the
80-degree angle.

00:03:46.840 --> 00:03:48.510
So over here, the
80-degree angle

00:03:48.510 --> 00:03:51.360
is going to be M, the one that
we don't have any label for.

00:03:51.360 --> 00:03:53.060
It's kind of the
other side-- it's

00:03:53.060 --> 00:03:56.960
the thing that shares the 7
length side right over here.

00:03:56.960 --> 00:04:03.239
So then we want to go to
N, then M-- sorry, NM--

00:04:03.239 --> 00:04:04.780
and then finish up
the triangle in O.

00:04:04.780 --> 00:04:06.226
And I want to
really stress this,

00:04:06.226 --> 00:04:08.600
that we have to make sure we
get the order of these right

00:04:08.600 --> 00:04:11.110
because then we're referring
to-- we're not showing

00:04:11.110 --> 00:04:13.490
the corresponding
vertices in each triangle.

00:04:13.490 --> 00:04:16.990
Now we see vertex
A, or point A, maps

00:04:16.990 --> 00:04:19.940
to point N on this
congruent triangle.

00:04:19.940 --> 00:04:23.600
Vertex B maps to
point M. And so you

00:04:23.600 --> 00:04:28.540
can say, look, the length
of AB is congruent to NM.

00:04:28.540 --> 00:04:29.546
So it all matches up.

00:04:29.546 --> 00:04:30.920
And we can say
that these two are

00:04:30.920 --> 00:04:36.500
congruent by angle,
angle, side, by AAS.

00:04:36.500 --> 00:04:39.570
So we did this one, this
one right over here,

00:04:39.570 --> 00:04:43.720
is congruent to this
one right over there.

00:04:43.720 --> 00:04:46.390
And now let's look at
these two characters.

00:04:46.390 --> 00:04:49.600
So here we have an angle, 40
degrees, a side in between,

00:04:49.600 --> 00:04:50.650
and then another angle.

00:04:50.650 --> 00:04:53.290
So it looks like ASA is
going to be involved.

00:04:53.290 --> 00:04:55.540
We look at this one
right over here.

00:04:55.540 --> 00:04:59.672
We have 40 degrees, 40
degrees, 7, and then 60.

00:04:59.672 --> 00:05:02.130
You might say, wait, here are
the 40 degrees on the bottom.

00:05:02.130 --> 00:05:02.775
Then here it's on the top.

00:05:02.775 --> 00:05:04.520
But remember, things
can be congruent

00:05:04.520 --> 00:05:06.970
if you can flip them-- if
you could flip them, rotate

00:05:06.970 --> 00:05:08.430
them, shift them, whatever.

00:05:08.430 --> 00:05:09.970
So if you flip
this guy over, you

00:05:09.970 --> 00:05:11.250
will get this one over here.

00:05:11.250 --> 00:05:12.875
And that would not
have happened if you

00:05:12.875 --> 00:05:15.240
had flipped this one to
get this one over here.

00:05:15.240 --> 00:05:19.010
So you see these two by--
let me just make it clear--

00:05:19.010 --> 00:05:21.810
you have this 60-degree angle
is congruent to this 60-degree

00:05:21.810 --> 00:05:22.580
angle.

00:05:22.580 --> 00:05:24.830
You have this side
of length 7 is

00:05:24.830 --> 00:05:26.740
congruent to this
side of length 7.

00:05:26.740 --> 00:05:28.810
And then you have
the 40-degree angle

00:05:28.810 --> 00:05:31.100
is congruent to this
40-degree angle.

00:05:31.100 --> 00:05:33.070
So once again,
these two characters

00:05:33.070 --> 00:05:35.040
are congruent to each other.

00:05:35.040 --> 00:05:38.600
And we can write-- I'll
write it right over here-- we

00:05:38.600 --> 00:05:46.900
can say triangle DEF is
congruent to triangle--

00:05:46.900 --> 00:05:48.500
and here we have to
be careful again.

00:05:48.500 --> 00:05:51.880
D, point D, is the vertex
for the 60-degree side.

00:05:51.880 --> 00:05:54.610
So I'm going to start at H,
which is the vertex of the 60--

00:05:54.610 --> 00:05:58.020
degree side over here-- is
congruent to triangle H.

00:05:58.020 --> 00:06:00.200
And then we went
from D to E. E is

00:06:00.200 --> 00:06:03.390
the vertex on the 40-degree
side, the other vertex that

00:06:03.390 --> 00:06:06.480
shares the 7 length
segment right over here.

00:06:06.480 --> 00:06:11.300
So we want to go
from H to G, HGI,

00:06:11.300 --> 00:06:13.400
and we know that from
angle, side, angle.

00:06:16.710 --> 00:06:18.840
And so that gives us that
that character right over

00:06:18.840 --> 00:06:22.940
there is congruent to this
character right over here.

00:06:22.940 --> 00:06:25.340
And then finally, we're left
with this poor, poor chap.

00:06:25.340 --> 00:06:28.439
And it looks like it is not
congruent to any of them.

00:06:28.439 --> 00:06:30.480
It is tempting to try to
match it up to this one,

00:06:30.480 --> 00:06:32.710
especially because the
angles here are on the bottom

00:06:32.710 --> 00:06:34.668
and you have the 7 side
over here-- angles here

00:06:34.668 --> 00:06:36.697
on the bottom and
the 7 side over here.

00:06:36.697 --> 00:06:39.030
But it doesn't match up,
because the order of the angles

00:06:39.030 --> 00:06:39.590
aren't the same.

00:06:39.590 --> 00:06:41.410
You don't have the same
corresponding angles.

00:06:41.410 --> 00:06:43.201
If you try to do this
little exercise where

00:06:43.201 --> 00:06:44.660
you map everything
to each other,

00:06:44.660 --> 00:06:46.576
you wouldn't be able to
do it right over here.

00:06:46.576 --> 00:06:49.030
And this over here-- it might
have been a trick question

00:06:49.030 --> 00:06:50.738
where maybe if you
did the math-- if this

00:06:50.738 --> 00:06:52.620
was like a 40 or a
60-degree angle, then

00:06:52.620 --> 00:06:54.485
maybe you could
have matched this

00:06:54.485 --> 00:06:56.860
to some of the other triangles
or maybe even some of them

00:06:56.860 --> 00:06:57.830
to each other.

00:06:57.830 --> 00:07:01.220
But this last angle, in all
of these cases-- 40 plus 60

00:07:01.220 --> 00:07:02.697
is 100.

00:07:02.697 --> 00:07:04.780
This is going to be an
80-degree angle right over.

00:07:04.780 --> 00:07:05.905
They have to add up to 180.

00:07:05.905 --> 00:07:09.210
This is an 80-degree angle.

00:07:09.210 --> 00:07:12.140
If this ended up, by the math,
being a 40 or 60-degree angle,

00:07:12.140 --> 00:07:14.390
then it could have been a
little bit more interesting.

00:07:14.390 --> 00:07:16.600
There might have been
other congruent pairs.

00:07:16.600 --> 00:07:19.360
But this is an 80-degree
angle in every case.

00:07:19.360 --> 00:07:20.650
The other angle is 80 degrees.

00:07:20.650 --> 00:07:23.680
So this is just a lone--
unfortunately for him,

00:07:23.680 --> 00:07:26.994
he is not able to find
a congruent companion.


Sum of the exterior angles of convex polygon

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:01.800
Several videos ago,

00:00:01.800 --> 00:00:03.770
I had a figure that looked something like this.

00:00:03.770 --> 00:00:05.750
I believe it was a pentagon or a hexagon.

00:00:05.750 --> 00:00:07.940
And what we had to do is figure out the sum

00:00:07.940 --> 00:00:11.040
of the particular exterior angles of the hexagon.

00:00:11.040 --> 00:00:12.570
So it would've been this angle,

00:00:12.570 --> 00:00:17.570
we should call A, this angle B, C, D, and E.

00:00:17.880 --> 00:00:19.527
And the way that we did it the last time, we said,

00:00:19.527 --> 00:00:21.927
"Well, A is going to be 180 degrees

00:00:21.927 --> 00:00:25.090
"minus the interior angle that is supplementary to A."

00:00:25.090 --> 00:00:27.190
And then we did that for each of the angles.

00:00:27.190 --> 00:00:28.900
And then we figured out we were able to

00:00:28.900 --> 00:00:30.150
algebraically manipulate it.

00:00:30.150 --> 00:00:31.560
We were able to figure out what the sum

00:00:31.560 --> 00:00:32.960
of the interior angles were

00:00:32.960 --> 00:00:35.060
using dividing it up into triangles,

00:00:35.060 --> 00:00:37.120
and then use that to figure out the exterior angles.

00:00:37.120 --> 00:00:39.170
And it was a bit of an involved process.

00:00:39.170 --> 00:00:40.410
What I want to show you in this video

00:00:40.410 --> 00:00:42.740
is there's actually a pretty simple and elegant way

00:00:42.740 --> 00:00:46.590
to figure out the sum of these particular external angles,

00:00:46.590 --> 00:00:49.240
exterior angles I should say, of this polygon.

00:00:49.240 --> 00:00:52.440
And it actually works for any convex polygons

00:00:52.440 --> 00:00:55.280
when you're picking these particular external,

00:00:55.280 --> 00:00:58.860
these particular exterior angles, I should say.

00:00:58.860 --> 00:01:01.170
And so the way to think about it is

00:01:01.170 --> 00:01:03.770
you can just redraw the angles.

00:01:03.770 --> 00:01:06.100
So let's just draw each of them.

00:01:06.100 --> 00:01:09.310
So let me draw this angle right over here.

00:01:09.310 --> 00:01:11.670
We could call it angle A or maybe the measure of this angle

00:01:11.670 --> 00:01:12.840
is A, either way.

00:01:12.840 --> 00:01:14.650
Let me draw it right over here.

00:01:14.650 --> 00:01:15.483
So it's going to be,

00:01:15.483 --> 00:01:17.450
this is going to be a congruent angle, right over here.

00:01:17.450 --> 00:01:19.770
It's going to have a measure of A.

00:01:19.770 --> 00:01:24.450
Now let me draw angle B, angle B.

00:01:24.450 --> 00:01:26.790
And I'm going to draw adjacent to angle A.

00:01:26.790 --> 00:01:28.480
And what you could do is think about it.

00:01:28.480 --> 00:01:30.950
Maybe if we drew a line right over here,

00:01:30.950 --> 00:01:32.340
if we drew a line right over here

00:01:32.340 --> 00:01:34.920
that was parallel to this line, then the measure

00:01:34.920 --> 00:01:37.630
of this angle right over here would also be B,

00:01:37.630 --> 00:01:39.890
because this obviously is a straight line.

00:01:39.890 --> 00:01:41.110
It would be like a transversal.

00:01:41.110 --> 00:01:42.760
These are corresponding angles.

00:01:42.760 --> 00:01:44.860
So if we wanted to draw the adjacent angle

00:01:44.860 --> 00:01:47.900
be adjacent to A, you could do it like that

00:01:47.900 --> 00:01:50.950
or the whatever angle this is, its measure is B.

00:01:50.950 --> 00:01:53.370
Then now it's adjacent to A,

00:01:53.370 --> 00:01:55.580
and now let's draw the same thing for C.

00:01:55.580 --> 00:01:58.730
We could draw a parallel line to that right over here.

00:01:58.730 --> 00:02:00.630
And then this angle would also be C.

00:02:00.630 --> 00:02:02.650
And if we want it to be adjacent to that,

00:02:02.650 --> 00:02:04.080
we could draw it right over here.

00:02:04.080 --> 00:02:05.760
So that angle is C.

00:02:05.760 --> 00:02:08.590
So C would look something like this.

00:02:08.590 --> 00:02:10.570
C would look something like that.

00:02:10.570 --> 00:02:12.310
Then we can move on to D.

00:02:12.310 --> 00:02:15.150
Once again, let me do that in a different color.

00:02:15.150 --> 00:02:16.820
You could do D.

00:02:16.820 --> 00:02:19.260
D could be right over here,

00:02:19.260 --> 00:02:21.440
or you could shift it down over here to look like that.

00:02:21.440 --> 00:02:24.040
Or you could shift it over here to look like that.

00:02:24.040 --> 00:02:25.780
If we just kept thinking about parallel...

00:02:25.780 --> 00:02:28.270
If all of these lines were parallel to each other,

00:02:28.270 --> 00:02:31.020
so let's just draw D like this.

00:02:31.020 --> 00:02:32.993
Let's just draw D like this.

00:02:33.980 --> 00:02:34.880
D like this.

00:02:34.880 --> 00:02:37.720
So this line once again's gonna be parallel to that line.

00:02:37.720 --> 00:02:40.050
And then finally, you have E.

00:02:40.050 --> 00:02:41.450
Finally, you have angle E.

00:02:41.450 --> 00:02:43.450
And once again, you could draw a line.

00:02:43.450 --> 00:02:45.570
You could draw a line that is parallel to this

00:02:45.570 --> 00:02:47.090
right over here.

00:02:47.090 --> 00:02:48.660
Right over here, and this right over here

00:02:48.660 --> 00:02:52.290
would be angle E, or you can draw it right over here.

00:02:52.290 --> 00:02:53.870
You would draw it right over here.

00:02:53.870 --> 00:02:55.710
And when you see it drawn this way,

00:02:55.710 --> 00:02:58.480
it's clear that when you add up the measure,

00:02:58.480 --> 00:03:02.660
this angle A, B, C, D, and E, you're going all the way

00:03:02.660 --> 00:03:04.150
around the circle.

00:03:04.150 --> 00:03:05.406
Either way, you could be going...

00:03:05.406 --> 00:03:07.160
You could be going clockwise,

00:03:07.160 --> 00:03:08.850
or you could be going counter-clockwise,

00:03:08.850 --> 00:03:11.170
but you're going all the way around the circle.

00:03:11.170 --> 00:03:13.810
And so the sum of these angles are just going to be...

00:03:13.810 --> 00:03:18.810
So A plus B, plus C, plus D, plus E

00:03:18.940 --> 00:03:22.310
is just going to be 360 degrees.

00:03:22.310 --> 00:03:24.100
And this will actually work as I said,

00:03:24.100 --> 00:03:25.720
for any convex polygon.

00:03:25.720 --> 00:03:27.040
When I say convex polygon,

00:03:27.040 --> 00:03:30.330
I mean one that's not dented inwards, one that's kind of...

00:03:30.330 --> 00:03:32.330
So just to be clear, what I'm talking about...

00:03:32.330 --> 00:03:37.330
It would work for any polygon that is kind of...

00:03:37.760 --> 00:03:38.740
I don't want to say regular.

00:03:38.740 --> 00:03:41.330
Regular means it has the same sides and same angles,

00:03:41.330 --> 00:03:42.420
but it's not dented.

00:03:42.420 --> 00:03:46.980
So this is a convex polygon.

00:03:46.980 --> 00:03:49.620
This right here is a concave polygon.

00:03:49.620 --> 00:03:51.640
So let me draw it this way.

00:03:51.640 --> 00:03:54.720
So this right over here would be a concave,

00:03:54.720 --> 00:03:57.120
would be a concave polygon.

00:03:57.120 --> 00:04:00.590
I'm gonna draw it as a having the same number of sides.

00:04:00.590 --> 00:04:04.160
So I just kind of dented these two sides right over there.

00:04:04.160 --> 00:04:05.610
And did I do that right?

00:04:05.610 --> 00:04:06.443
Let me see.

00:04:06.443 --> 00:04:07.750
Let me do it the same number of sides.

00:04:07.750 --> 00:04:11.930
So I want to do that, that, that, that,

00:04:11.930 --> 00:04:15.080
and then I know that's the same side over there.

00:04:15.080 --> 00:04:15.913
Let me draw it like that.

00:04:15.913 --> 00:04:17.780
And then like that.

00:04:17.780 --> 00:04:20.810
This has one, two, three, four, five, six sides.

00:04:20.810 --> 00:04:24.850
This has one, two, three, four, five, six sides.

00:04:24.850 --> 00:04:27.060
This is concave.

00:04:27.060 --> 00:04:29.400
Sorry, this is convex.

00:04:29.400 --> 00:04:31.820
This is a convex polygon.

00:04:31.820 --> 00:04:34.170
This is a concave polygon.

00:04:34.170 --> 00:04:37.460
And the way I remember it is kind of caved inwards.

00:04:37.460 --> 00:04:39.480
And so what we just did would apply to any.

00:04:39.480 --> 00:04:41.720
If we're trying to find these particular external,

00:04:41.720 --> 00:04:44.800
exterior angles of any convex polygon,

00:04:44.800 --> 00:04:47.380
I afraid, I apologize ahead of time

00:04:47.380 --> 00:04:48.420
if I've confused them all,

00:04:48.420 --> 00:04:50.070
because I have a feeling that I might've.

00:04:50.070 --> 00:04:52.670
This applies to any convex polygon.

00:04:52.670 --> 00:04:56.930
And so once again, if you take this angle

00:04:56.930 --> 00:05:00.500
and add it to this angle, and add it to this angle,

00:05:00.500 --> 00:05:03.500
add it to this angle, add it to that angle,

00:05:03.500 --> 00:05:04.820
and add it to that angle.

00:05:04.820 --> 00:05:06.790
And I'm not implying that they're all going to be the same.

00:05:06.790 --> 00:05:08.070
I just drew it that way.

00:05:08.070 --> 00:05:09.590
I could show you that they are different angles.

00:05:09.590 --> 00:05:11.300
So I could say that one in green

00:05:11.300 --> 00:05:13.740
and that one in some other color, I think you get the idea.

00:05:13.740 --> 00:05:15.100
They can all be different,

00:05:15.100 --> 00:05:17.050
but when you if you shift the angles like this

00:05:17.050 --> 00:05:18.840
you'll see that they just go around the circle.

00:05:18.840 --> 00:05:23.303
So once again, they'll just add up to 360 degrees.


Sum of interior angles of a polygon

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.240
We already know that the
sum of the interior angles

00:00:03.240 --> 00:00:06.560
of a triangle add
up to 180 degrees.

00:00:06.560 --> 00:00:08.415
So if the measure
of this angle is a,

00:00:08.415 --> 00:00:10.460
the measure of this
angle over here is b,

00:00:10.460 --> 00:00:11.940
and the measure of
this angle is c,

00:00:11.940 --> 00:00:18.030
we know that a plus b plus
c is equal to 180 degrees.

00:00:18.030 --> 00:00:19.800
But what happens
when we have polygons

00:00:19.800 --> 00:00:21.300
with more than three sides?

00:00:21.300 --> 00:00:23.460
So let's try the
case where we have

00:00:23.460 --> 00:00:26.300
a four-sided polygon--
a quadrilateral.

00:00:26.300 --> 00:00:28.120
And I am going to
make it irregular just

00:00:28.120 --> 00:00:30.600
to show that whatever we
do here it probably applies

00:00:30.600 --> 00:00:32.270
to any quadrilateral
with four sides.

00:00:32.270 --> 00:00:35.282
Not just things that have right
angles, and parallel lines,

00:00:35.282 --> 00:00:35.990
and all the rest.

00:00:35.990 --> 00:00:36.910
Actually, that
looks a little bit

00:00:36.910 --> 00:00:38.250
too close to being parallel.

00:00:38.250 --> 00:00:39.416
So let me draw it like this.

00:00:41.475 --> 00:00:43.350
So the way you can think
about it with a four

00:00:43.350 --> 00:00:45.090
sided quadrilateral,
is well we already

00:00:45.090 --> 00:00:47.970
know about this-- the measures
of the interior angles

00:00:47.970 --> 00:00:49.680
of a triangle add up to 180.

00:00:49.680 --> 00:00:51.820
So maybe we can divide
this into two triangles.

00:00:51.820 --> 00:00:53.830
So from this point
right over here,

00:00:53.830 --> 00:00:57.620
if we draw a line
like this, we've

00:00:57.620 --> 00:00:59.970
divided it into two triangles.

00:00:59.970 --> 00:01:02.550
And so if the measure this
angle is a, measure of this

00:01:02.550 --> 00:01:04.580
is b, measure of
that is c, we know

00:01:04.580 --> 00:01:09.540
that a plus b plus c is
equal to 180 degrees.

00:01:09.540 --> 00:01:13.310
And then if we call this over
here x, this over here y,

00:01:13.310 --> 00:01:15.530
and that z, those are the
measures of those angles.

00:01:15.530 --> 00:01:21.232
We know that x plus y plus
z is equal to 180 degrees.

00:01:21.232 --> 00:01:23.190
And so if we want the
measure of the sum of all

00:01:23.190 --> 00:01:26.830
of the interior angles,
all of the interior angles

00:01:26.830 --> 00:01:31.050
are going to be
b plus z-- that's

00:01:31.050 --> 00:01:33.270
two of the interior
angles of this polygon--

00:01:33.270 --> 00:01:39.000
plus this angle, which is
just going to be a plus x.

00:01:39.000 --> 00:01:40.860
a plus x is that whole angle.

00:01:40.860 --> 00:01:43.040
The whole angle for
the quadrilateral.

00:01:43.040 --> 00:01:46.927
Plus this whole angle, which
is going to be c plus y.

00:01:49.570 --> 00:01:52.185
And we already know a plus
b plus c is 180 degrees.

00:01:57.420 --> 00:02:03.370
And we know that z plus x plus
y is equal to 180 degrees.

00:02:03.370 --> 00:02:08.322
So plus 180 degrees, which
is equal to 360 degrees.

00:02:08.322 --> 00:02:10.030
So I think you see
the general idea here.

00:02:10.030 --> 00:02:12.680
We just have to figure out how
many triangles we can divide

00:02:12.680 --> 00:02:16.760
something into, and then we
just multiply by 180 degrees

00:02:16.760 --> 00:02:20.160
since each of those triangles
will have 180 degrees.

00:02:20.160 --> 00:02:22.210
Let's do one more
particular example.

00:02:22.210 --> 00:02:24.461
And then we'll try to do a
general version where we're

00:02:24.461 --> 00:02:26.585
just trying to figure out
how many triangles can we

00:02:26.585 --> 00:02:27.630
fit into that thing.

00:02:27.630 --> 00:02:29.760
So let me draw an
irregular pentagon.

00:02:29.760 --> 00:02:36.070
So one, two, three, four, five.

00:02:36.070 --> 00:02:38.820
So it looks like a little bit
of a sideways house there.

00:02:38.820 --> 00:02:41.260
Once again, we can draw
our triangles inside

00:02:41.260 --> 00:02:42.640
of this pentagon.

00:02:42.640 --> 00:02:44.640
So that would be
one triangle there.

00:02:44.640 --> 00:02:46.020
That would be another triangle.

00:02:46.020 --> 00:02:49.740
So I'm able to draw three
non-overlapping triangles that

00:02:49.740 --> 00:02:51.350
perfectly cover this pentagon.

00:02:51.350 --> 00:02:55.122
This is one triangle, the other
triangle, and the other one.

00:02:55.122 --> 00:02:57.080
And we know each of those
will have 180 degrees

00:02:57.080 --> 00:02:58.889
if we take the sum
of their angles.

00:02:58.889 --> 00:03:01.430
And we also know that the sum
of all of those interior angles

00:03:01.430 --> 00:03:04.050
are equal to the sum
of the interior angles

00:03:04.050 --> 00:03:05.240
of the polygon as a whole.

00:03:05.240 --> 00:03:07.540
And to see that, clearly,
this interior angle

00:03:07.540 --> 00:03:09.850
is one of the angles
of the polygon.

00:03:09.850 --> 00:03:11.290
This is as well.

00:03:11.290 --> 00:03:14.587
But when you take the sum
of this one and this one,

00:03:14.587 --> 00:03:16.670
then you're going to get
that whole interior angle

00:03:16.670 --> 00:03:17.750
of the polygon.

00:03:17.750 --> 00:03:19.875
And when you take the sum
of that one and that one,

00:03:19.875 --> 00:03:21.250
you get that entire one.

00:03:21.250 --> 00:03:23.270
And then when you take
the sum of that one

00:03:23.270 --> 00:03:26.972
plus that one plus that one, you
get that entire interior angle.

00:03:26.972 --> 00:03:29.430
So if you take the sum of all
of the interior angles of all

00:03:29.430 --> 00:03:31.430
of these triangles, you're
actually just finding

00:03:31.430 --> 00:03:34.810
the sum of all of the interior
angles of the polygon.

00:03:34.810 --> 00:03:37.990
So in this case, you have
one, two, three triangles.

00:03:37.990 --> 00:03:41.980
So three times 180
degrees is equal to what?

00:03:41.980 --> 00:03:46.360
300 plus 240 is
equal to 540 degrees.

00:03:46.360 --> 00:03:47.940
Now let's generalize it.

00:03:47.940 --> 00:03:49.520
And to generalize
it, let's realize

00:03:49.520 --> 00:03:52.280
that just to get our
first two triangles,

00:03:52.280 --> 00:03:54.160
we have to use up four sides.

00:03:54.160 --> 00:03:58.250
We have to use up all the four
sides in this quadrilateral.

00:03:58.250 --> 00:04:01.161
We had to use up four
of the five sides--

00:04:01.161 --> 00:04:02.410
right here-- in this pentagon.

00:04:02.410 --> 00:04:04.480
One, two, and then three, four.

00:04:04.480 --> 00:04:08.550
So four sides give
you two triangles.

00:04:08.550 --> 00:04:10.970
And it seems like, maybe,
every incremental side

00:04:10.970 --> 00:04:13.470
you have after that, you can
get another triangle out of it.

00:04:13.470 --> 00:04:15.130
Let's experiment with a hexagon.

00:04:15.130 --> 00:04:17.700
And I'm just going to try to
see how many triangles I get out

00:04:17.700 --> 00:04:18.200
of it.

00:04:18.200 --> 00:04:23.283
So one, two, three,
four, five, six sides.

00:04:27.260 --> 00:04:30.140
I get one triangle out
of these two sides.

00:04:30.140 --> 00:04:32.980
One, two sides of
the actual hexagon.

00:04:32.980 --> 00:04:35.050
I can get another
triangle out of these two

00:04:35.050 --> 00:04:38.450
sides of the actual hexagon.

00:04:38.450 --> 00:04:40.920
And it looks like I can
get another triangle out

00:04:40.920 --> 00:04:42.520
of each of the remaining sides.

00:04:42.520 --> 00:04:44.010
So one out of that one.

00:04:44.010 --> 00:04:46.860
And then one out of that
one, right over there.

00:04:46.860 --> 00:04:51.060
So in general, it
seems like-- let's say.

00:04:51.060 --> 00:04:55.110
So let's say that
I have s sides.

00:04:55.110 --> 00:04:56.565
s-sided polygon.

00:05:00.786 --> 00:05:02.160
And I'll just
assume-- we already

00:05:02.160 --> 00:05:07.230
saw the case for four sides,
five sides, or six sides.

00:05:07.230 --> 00:05:10.740
So we can assume that s
is greater than 4 sides.

00:05:10.740 --> 00:05:12.240
Let's say I have
an s-sided polygon,

00:05:12.240 --> 00:05:15.240
and I want to figure out how
many non-overlapping triangles

00:05:15.240 --> 00:05:16.900
will perfectly
cover that polygon.

00:05:16.900 --> 00:05:18.234
How many can I fit inside of it?

00:05:18.234 --> 00:05:20.566
And then I just have to
multiply the number of triangles

00:05:20.566 --> 00:05:22.500
times 180 degrees
to figure out what

00:05:22.500 --> 00:05:25.690
are the sum of the interior
angles of that polygon.

00:05:25.690 --> 00:05:28.600
So let's figure out
the number of triangles

00:05:28.600 --> 00:05:31.170
as a function of
the number of sides.

00:05:31.170 --> 00:05:34.320
So once again, four
of the sides are

00:05:34.320 --> 00:05:36.060
going to be used to
make two triangles.

00:05:36.060 --> 00:05:38.120
So those two sides
right over there.

00:05:38.120 --> 00:05:41.172
And then we have two
sides right over there.

00:05:41.172 --> 00:05:43.130
I can draw one triangle
over-- and I'm not even

00:05:43.130 --> 00:05:45.294
going to talk about
what happens on the rest

00:05:45.294 --> 00:05:46.460
of the sides of the polygon.

00:05:46.460 --> 00:05:48.918
You could imagine putting a
big black piece of construction

00:05:48.918 --> 00:05:49.470
paper.

00:05:49.470 --> 00:05:50.959
There might be other sides here.

00:05:50.959 --> 00:05:53.000
I'm not going to even
worry about them right now.

00:05:53.000 --> 00:05:54.610
So out of these two
sides I can draw

00:05:54.610 --> 00:05:56.420
one triangle, just like that.

00:05:56.420 --> 00:05:58.980
Out of these two sides, I can
draw another triangle right

00:05:58.980 --> 00:05:59.640
over there.

00:05:59.640 --> 00:06:05.550
So four sides used
for two triangles.

00:06:08.340 --> 00:06:11.900
And then, no matter how many
sides I have left over-- so

00:06:11.900 --> 00:06:14.280
I've already used four of
the sides, but after that,

00:06:14.280 --> 00:06:16.900
if I have all sorts
of craziness here.

00:06:16.900 --> 00:06:19.300
I could have all sorts
of craziness here.

00:06:19.300 --> 00:06:21.420
Let me draw it a little
bit neater than that.

00:06:21.420 --> 00:06:26.950
So I could have all sorts of
craziness right over here.

00:06:26.950 --> 00:06:28.820
It looks like every
other incremental side

00:06:28.820 --> 00:06:30.380
I can get another
triangle out of it.

00:06:30.380 --> 00:06:33.060
So that's one triangle out
of there, one triangle out

00:06:33.060 --> 00:06:35.500
of that side, one
triangle out of that side,

00:06:35.500 --> 00:06:38.220
one triangle out of that side,
and then one triangle out

00:06:38.220 --> 00:06:39.210
of this side.

00:06:39.210 --> 00:06:41.290
So for example, this
figure that I've drawn

00:06:41.290 --> 00:06:46.000
is a very irregular-- one, two,
three, four, five, six, seven,

00:06:46.000 --> 00:06:47.970
eight, nine, 10.

00:06:47.970 --> 00:06:48.620
Is that right?

00:06:48.620 --> 00:06:52.870
One, two, three, four, five,
six, seven, eight, nine, 10.

00:06:52.870 --> 00:06:54.210
It is a decagon.

00:06:54.210 --> 00:06:57.050
And in this decagon,
four of the sides

00:06:57.050 --> 00:06:58.220
were used for two triangles.

00:06:58.220 --> 00:07:00.910
So I got two triangles
out of four of the sides.

00:07:00.910 --> 00:07:02.820
And out of the other
six sides I was

00:07:02.820 --> 00:07:04.120
able to get a triangle each.

00:07:04.120 --> 00:07:05.440
These are six.

00:07:05.440 --> 00:07:08.175
This is one, two,
three, four, five.

00:07:08.175 --> 00:07:10.633
Actually, let me make sure I'm
counting the number of sides

00:07:10.633 --> 00:07:11.540
right.

00:07:11.540 --> 00:07:21.909
So I have one, two, three, four,
five, six, seven, eight, nine,

00:07:21.909 --> 00:07:22.747
10.

00:07:22.747 --> 00:07:23.580
So let me make sure.

00:07:23.580 --> 00:07:27.691
Did I count-- am I just
not seeing something?

00:07:27.691 --> 00:07:28.190
Oh, I see.

00:07:28.190 --> 00:07:30.410
I actually didn't-- I have to
draw another line right over

00:07:30.410 --> 00:07:30.770
here.

00:07:30.770 --> 00:07:32.450
These are two different
sides, and so I

00:07:32.450 --> 00:07:33.740
have to draw another
line right over here.

00:07:33.740 --> 00:07:36.319
I can get another triangle
out of that right over there.

00:07:36.319 --> 00:07:37.360
And so there you have it.

00:07:37.360 --> 00:07:39.960
I have these two triangles
out of four sides.

00:07:39.960 --> 00:07:41.800
And out of the other
six remaining sides

00:07:41.800 --> 00:07:43.250
I get a triangle each.

00:07:43.250 --> 00:07:45.110
So plus six triangles.

00:07:45.110 --> 00:07:47.610
I got a total of
eight triangles.

00:07:47.610 --> 00:07:51.700
And so we can generally
think about it.

00:07:51.700 --> 00:07:55.510
The first four, sides we're
going to get two triangles.

00:07:55.510 --> 00:07:57.170
So let me write this down.

00:07:57.170 --> 00:08:01.630
So our number of triangles
is going to be equal to 2.

00:08:01.630 --> 00:08:04.370
And then, I've already
used four sides.

00:08:04.370 --> 00:08:07.070
So the remaining sides
I get a triangle each.

00:08:07.070 --> 00:08:10.440
So the remaining sides
are going to be s minus 4.

00:08:10.440 --> 00:08:13.730
So the number of triangles are
going to be 2 plus s minus 4.

00:08:13.730 --> 00:08:17.280
2 plus s minus 4
is just s minus 2.

00:08:17.280 --> 00:08:21.910
So if I have an
s-sided polygon, I

00:08:21.910 --> 00:08:26.110
can get s minus 2 triangles that
perfectly cover that polygon

00:08:26.110 --> 00:08:28.830
and that don't overlap with
each other, which tells us

00:08:28.830 --> 00:08:32.549
that an s-sided polygon, if
it has s minus 2 triangles,

00:08:32.549 --> 00:08:35.809
that the interior angles
in it are going to be

00:08:35.809 --> 00:08:40.559
s minus 2 times 180 degrees.

00:08:40.559 --> 00:08:42.309
Which is a pretty cool result.

00:08:42.309 --> 00:08:47.540
So if someone told you that they
had a 102-sided polygon-- so

00:08:47.540 --> 00:08:49.680
s is equal to 102 sides.

00:08:49.680 --> 00:08:51.990
You can say, OK, the
number of interior angles

00:08:51.990 --> 00:08:55.140
are going to be 102 minus 2.

00:08:55.140 --> 00:08:58.730
So it's going to be 100
times 180 degrees, which

00:08:58.730 --> 00:09:02.850
is equal to 180 with two
more zeroes behind it.

00:09:02.850 --> 00:09:06.620
So it'd be 18,000 degrees
for the interior angles

00:09:06.620 --> 00:09:09.930
of a 102-sided polygon.


Two column proof showing segments are perpendicular

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:02.150
What I want to do
in this video is

00:00:02.150 --> 00:00:10.000
I want to prove that segment
AC is perpendicular to segment

00:00:10.000 --> 00:00:12.800
DB based on the
information that we

00:00:12.800 --> 00:00:15.300
have in this diagram over here.

00:00:15.300 --> 00:00:17.850
That this side has the
same length as that side,

00:00:17.850 --> 00:00:21.040
this side has the same
length as that side.

00:00:21.040 --> 00:00:23.940
And I'll give you a hint,
we're going to use one or more

00:00:23.940 --> 00:00:27.130
of our congruence postulates.

00:00:27.130 --> 00:00:29.850
I'll just stick with calling
them postulates from now on.

00:00:29.850 --> 00:00:31.770
And so the ones that
we know-- so let me

00:00:31.770 --> 00:00:33.220
draw a little line here.

00:00:33.220 --> 00:00:34.460
This is kind of our tool kit.

00:00:34.460 --> 00:00:36.750
We have the side
side side postulate,

00:00:36.750 --> 00:00:38.930
if the three sides
are congruent,

00:00:38.930 --> 00:00:40.900
then the two triangles
are congruent.

00:00:40.900 --> 00:00:44.960
We have side angle side, two
sides and the angle in between

00:00:44.960 --> 00:00:47.780
are congruent, then the two
triangles are congruent.

00:00:47.780 --> 00:00:51.590
We have ASA, two angles
with a side in between.

00:00:51.590 --> 00:00:54.660
And then we have AAS, two
angles and then a side.

00:00:54.660 --> 00:00:56.660
So any of these things
we've established-- these

00:00:56.660 --> 00:00:58.490
are our postulates--
we're going to assume

00:00:58.490 --> 00:01:00.300
that they imply congruency.

00:01:00.300 --> 00:01:02.781
And I'm also going to do this
as what we call a two column

00:01:02.781 --> 00:01:03.280
proof.

00:01:03.280 --> 00:01:05.459
And you don't have to do
something as a two column

00:01:05.459 --> 00:01:06.958
proof, but this is
what you normally

00:01:06.958 --> 00:01:09.600
see in a normal
introductory geometry class.

00:01:09.600 --> 00:01:12.820
So I thought I would
expose you to it.

00:01:12.820 --> 00:01:16.267
It's a pretty basic idea is
that you make a statement,

00:01:16.267 --> 00:01:18.600
and you just have to give the
reason for your statement.

00:01:18.600 --> 00:01:20.516
Which is what we've been
doing with any proof,

00:01:20.516 --> 00:01:23.459
but we haven't always put
it in a very structured way.

00:01:23.459 --> 00:01:25.000
So I'm just going
to do it like this.

00:01:25.000 --> 00:01:28.480
I'll have two columns like that.

00:01:28.480 --> 00:01:32.500
And I'll have a
statement, and then I

00:01:32.500 --> 00:01:36.590
will give the reason
for the statement.

00:01:36.590 --> 00:01:39.360
And so the strategy that
I'm going to try to do

00:01:39.360 --> 00:01:42.310
is it looks like,
right off the bat,

00:01:42.310 --> 00:01:45.120
it seems like I can
prove the triangle CDA

00:01:45.120 --> 00:01:50.890
is congruent to triangle
CBA based on side side side.

00:01:50.890 --> 00:01:52.672
And that's a pretty
good starting point

00:01:52.672 --> 00:01:54.130
because once I can
base congruency,

00:01:54.130 --> 00:01:56.322
then I can start to
have angles be the same.

00:01:56.322 --> 00:01:57.780
And the reason why
I can do that is

00:01:57.780 --> 00:01:59.640
because this side is
the same as that side,

00:01:59.640 --> 00:02:02.497
this side the same as that side,
and they both share that side.

00:02:02.497 --> 00:02:04.580
But I don't want to just
do it verbally this time,

00:02:04.580 --> 00:02:07.990
I want to write it out properly
early in this two column proof.

00:02:07.990 --> 00:02:13.260
So we have CD, we had
the length of segment CD

00:02:13.260 --> 00:02:16.425
is equal to the length of CB.

00:02:16.425 --> 00:02:18.660
CD is equal to CB,
and that is given.

00:02:21.330 --> 00:02:25.310
So these two characters
have the same length.

00:02:25.310 --> 00:02:30.280
We also know that DA,
the length of segment DA,

00:02:30.280 --> 00:02:33.600
is the same as the
length of segment BA.

00:02:33.600 --> 00:02:41.000
So DA is equal to BA, that's
also given in the diagram.

00:02:41.000 --> 00:02:44.790
And then we also know
that CA is equal to CA,

00:02:44.790 --> 00:02:45.870
I guess we could say.

00:02:45.870 --> 00:02:48.930
So CA is equal to itself.

00:02:48.930 --> 00:02:51.460
And it's obviously
in both triangles.

00:02:51.460 --> 00:02:54.465
So this is also given, or
it's obvious from the diagram.

00:02:57.779 --> 00:02:58.570
It's a bit obvious.

00:02:58.570 --> 00:03:01.100
Both triangles share that side.

00:03:01.100 --> 00:03:03.850
So we have two triangles.

00:03:03.850 --> 00:03:06.680
Their corresponding sides
have the same length,

00:03:06.680 --> 00:03:08.890
and so we know that
they're congruent.

00:03:08.890 --> 00:03:17.970
So we know that triangle CDA
is congruent to triangle CBA.

00:03:21.620 --> 00:03:25.530
And we know that by the
side side side postulate

00:03:25.530 --> 00:03:27.267
and the statements
given up here.

00:03:27.267 --> 00:03:28.850
Actually, let me
number our statements

00:03:28.850 --> 00:03:33.730
just so we can refer back
to this 1, 2, 3, and 4.

00:03:33.730 --> 00:03:38.790
And so side side side
postulate and 1, 2, and 3--

00:03:38.790 --> 00:03:40.160
statements 1, 2, and 3.

00:03:40.160 --> 00:03:43.460
So statements 1, 2, and 3 and
the side side postulate let

00:03:43.460 --> 00:03:47.320
us know that these two
triangles are congruent.

00:03:47.320 --> 00:03:51.130
And then if these are congruent,
then we know, for example,

00:03:51.130 --> 00:03:55.860
we know that all of their
corresponding angles

00:03:55.860 --> 00:03:56.550
are equivalent.

00:03:56.550 --> 00:03:58.630
So for example,
this angle is going

00:03:58.630 --> 00:04:00.310
to be equal to that angle.

00:04:00.310 --> 00:04:02.226
So let's make that
statement right over there.

00:04:02.226 --> 00:04:04.040
We know that angle
DCE-- so this is

00:04:04.040 --> 00:04:09.460
going to be statement 5-- we
know that angle DCE, that's

00:04:09.460 --> 00:04:12.870
this angle right over here, is
going to have the same measure,

00:04:12.870 --> 00:04:14.440
we could even say
they're congruent.

00:04:14.440 --> 00:04:16.120
I'll say the
measure of angle DCE

00:04:16.120 --> 00:04:19.980
is going to be equal to
the measure of angle BCE.

00:04:23.210 --> 00:04:29.500
And this comes straight out
of statement 4, congruency,

00:04:29.500 --> 00:04:31.330
I could put it in parentheses.

00:04:31.330 --> 00:04:33.187
Congruency of those triangles.

00:04:33.187 --> 00:04:34.770
This implies straight,
because they're

00:04:34.770 --> 00:04:36.530
both part of this
larger triangle,

00:04:36.530 --> 00:04:38.420
they are the
corresponding angles,

00:04:38.420 --> 00:04:41.414
so they're going to have
the exact same measure.

00:04:41.414 --> 00:04:43.830
Now it seems like we could do
something pretty interesting

00:04:43.830 --> 00:04:46.520
with these two smaller
triangles at the top left

00:04:46.520 --> 00:04:49.350
and the top right of this,
looks like, a kite like figure.

00:04:49.350 --> 00:04:52.300
Because we have a side,
two corresponding sides

00:04:52.300 --> 00:04:55.490
are congruent, two corresponding
angles are congruent,

00:04:55.490 --> 00:04:57.490
and they have a side in common.

00:04:57.490 --> 00:04:59.760
They have this side in
common right over here.

00:04:59.760 --> 00:05:01.700
So let's first just
establish that they

00:05:01.700 --> 00:05:04.230
have this side in
common right over here.

00:05:04.230 --> 00:05:05.910
So I'll just write statement 6.

00:05:05.910 --> 00:05:09.760
We have CE, the measure or
the length of that line,

00:05:09.760 --> 00:05:10.800
is equal to itself.

00:05:10.800 --> 00:05:12.840
Once again, this
is just obvious.

00:05:12.840 --> 00:05:15.750
It's the same.

00:05:15.750 --> 00:05:18.780
Obvious from diagram
it's the same line.

00:05:18.780 --> 00:05:22.510
Obvious from diagram.

00:05:22.510 --> 00:05:24.760
But now we can use
that information.

00:05:24.760 --> 00:05:27.620
So we don't have three sides,
we haven't proven to ourselves

00:05:27.620 --> 00:05:29.350
that this side is the
same as this side,

00:05:29.350 --> 00:05:31.610
that DE has the
same length as EB.

00:05:31.610 --> 00:05:34.810
But we do have a side, an
angle between the sides,

00:05:34.810 --> 00:05:36.370
and then another side.

00:05:36.370 --> 00:05:40.430
And so this looks pretty
interesting for our side angle

00:05:40.430 --> 00:05:41.900
side postulate.

00:05:41.900 --> 00:05:46.180
And so we can say, by the
side angle side postulate,

00:05:46.180 --> 00:06:04.630
we can say that triangle DCE
is congruent to triangle BCE.

00:06:04.630 --> 00:06:07.254
And when I write the
labels for the triangles,

00:06:07.254 --> 00:06:08.920
I'm making sure that
I'm kind of putting

00:06:08.920 --> 00:06:10.170
the corresponding point.

00:06:10.170 --> 00:06:12.700
So I started at D, then
went to C, then to E.

00:06:12.700 --> 00:06:15.250
So the corresponding
I guess angle,

00:06:15.250 --> 00:06:17.960
or the corresponding point
or vertex I could say,

00:06:17.960 --> 00:06:19.900
for this triangle
right over here,

00:06:19.900 --> 00:06:23.704
is B. So if I start with D, I
start with B. C in the middle

00:06:23.704 --> 00:06:26.120
is the corresponding vertex
for either of these triangles,

00:06:26.120 --> 00:06:27.000
so I put it in the middle.

00:06:27.000 --> 00:06:29.333
And then they both go to E.
And that's just to make sure

00:06:29.333 --> 00:06:32.230
that we are specifying
what's corresponding to what.

00:06:32.230 --> 00:06:37.900
And we know this, we know this
is true, by side angle side.

00:06:37.900 --> 00:06:40.280
And the information
we got from-- so we

00:06:40.280 --> 00:06:43.180
got this side is established
that these two sides are

00:06:43.180 --> 00:06:45.270
congruent was from statement 1.

00:06:45.270 --> 00:06:49.730
Then that these angles are
congruent is from statement 5

00:06:49.730 --> 00:06:50.930
right over here.

00:06:50.930 --> 00:06:53.330
And then statement 6
gave us the other side.

00:06:53.330 --> 00:06:56.217
Statement 6, just like that.

00:06:56.217 --> 00:06:58.300
And if we know that these
triangles are congruent,

00:06:58.300 --> 00:07:00.920
that means that all of
their corresponding angles

00:07:00.920 --> 00:07:02.060
are congruent.

00:07:02.060 --> 00:07:04.980
So we know, for example, that
this angle right over here

00:07:04.980 --> 00:07:07.330
is going to be congruent
to that angle over there.

00:07:07.330 --> 00:07:08.850
So let's write that down.

00:07:08.850 --> 00:07:12.680
So we know, statement number
8, the measure of angle,

00:07:12.680 --> 00:07:19.578
let's call that DEC, is equal
to the measure of angle BEC.

00:07:23.370 --> 00:07:27.450
And this comes straight
from statement 7.

00:07:27.450 --> 00:07:29.830
Once again, they're congruent.

00:07:29.830 --> 00:07:32.690
Congruency.

00:07:32.690 --> 00:07:36.220
And then we also know--
we'll make statement 9.

00:07:36.220 --> 00:07:41.870
We also know that the measure
of angle DEC-- or maybe we

00:07:41.870 --> 00:07:43.660
should just write it this way.

00:07:43.660 --> 00:07:52.324
Angle DEC and angle
BEC are supplementary.

00:07:52.324 --> 00:07:53.740
They are supplementary,
and that's

00:07:53.740 --> 00:07:55.865
kind of-- you can just look
at that from inspection

00:07:55.865 --> 00:07:58.130
but I'll write a decent--
are supplementary.

00:07:58.130 --> 00:08:01.280
Supplementary, which
means they add up,

00:08:01.280 --> 00:08:05.010
their measures add
up, to 180 degrees.

00:08:05.010 --> 00:08:10.060
And we know that because
they are adjacent

00:08:10.060 --> 00:08:14.500
and outer sides
form straight angle.

00:08:20.950 --> 00:08:24.650
And then, the next step-- if we
know that these two angles are

00:08:24.650 --> 00:08:26.610
equal to each other,
and if we know

00:08:26.610 --> 00:08:29.250
that they are
complementary-- our next step

00:08:29.250 --> 00:08:30.790
means that we could
actually deduce

00:08:30.790 --> 00:08:32.990
that they must be 90 degrees.

00:08:32.990 --> 00:08:37.419
So 10, measure of
angle DEC equals

00:08:37.419 --> 00:08:44.205
measure of angle BEC,
which equals 90 degrees.

00:08:44.205 --> 00:08:46.830
And then for the reason it might
be a little bit more involved,

00:08:46.830 --> 00:08:49.140
we could put these two
statements together.

00:08:49.140 --> 00:08:52.810
So it would be
statements 8 and 9.

00:08:52.810 --> 00:08:56.550
And then statements 8
and 9 mean that DEC,

00:08:56.550 --> 00:09:01.670
so I could write this, measure
of angle DEC plus measure

00:09:01.670 --> 00:09:04.740
of angle-- actually,
let me just,

00:09:04.740 --> 00:09:07.490
since I don't want to do
too many steps all at once,

00:09:07.490 --> 00:09:09.910
let me just take it
little bit by little bit.

00:09:09.910 --> 00:09:12.560
So let me just do
it all like this.

00:09:12.560 --> 00:09:20.550
So let me say measure of angle
DEC plus measure of angle BEC

00:09:20.550 --> 00:09:22.100
is equal to 180.

00:09:22.100 --> 00:09:25.040
And this comes
straight from point 9,

00:09:25.040 --> 00:09:26.930
that they are supplementary.

00:09:26.930 --> 00:09:28.810
And then we could
say statement--

00:09:28.810 --> 00:09:32.040
I'm taking up a lot of
space now-- statement 11,

00:09:32.040 --> 00:09:35.140
we could say
measure of angle DEC

00:09:35.140 --> 00:09:41.180
plus measure of angle DEC
is equal to 180 degrees.

00:09:41.180 --> 00:09:43.390
And we know that
from statement 9.

00:09:43.390 --> 00:09:47.930
We know that from statement
9 and statement 8.

00:09:47.930 --> 00:09:49.430
We essentially just
took statement 9

00:09:49.430 --> 00:09:52.870
and substituted that
BEC, the measure of BEC,

00:09:52.870 --> 00:09:55.100
is the same as the
measure of DEC.

00:09:55.100 --> 00:09:57.900
And so then, if we
want statement 12,

00:09:57.900 --> 00:10:01.150
we could say
measure of angle DEC

00:10:01.150 --> 00:10:05.900
is equal to 90, which is equal
to the measure of angle BEC.

00:10:08.990 --> 00:10:13.570
And then, this comes once again
straight out of point number

00:10:13.570 --> 00:10:14.821
11 and 8.

00:10:14.821 --> 00:10:17.320
And what you could see, I'm
kind of taking a little bit more

00:10:17.320 --> 00:10:19.820
time, going a little bit more
granular through the steps.

00:10:19.820 --> 00:10:21.050
In some of the other proofs
I would've just said,

00:10:21.050 --> 00:10:23.260
oh, obviously this
implies this or that.

00:10:23.260 --> 00:10:24.200
And then we're done.

00:10:24.200 --> 00:10:27.377
Because if these
are 90 degrees-- so

00:10:27.377 --> 00:10:28.710
let me write the last statement.

00:10:28.710 --> 00:10:31.480
So statement 13, which is
what we wanted to prove.

00:10:31.480 --> 00:10:33.980
We wanted to prove that
AC is perpendicular to DB.

00:10:37.060 --> 00:10:42.890
So AC is perpendicular
to, what was it?

00:10:42.890 --> 00:10:47.530
AC is perpendicular
to segment DB,

00:10:47.530 --> 00:10:52.100
and it comes straight
out of point 12.

00:10:52.100 --> 00:10:52.760
And we're done.

00:10:52.760 --> 00:10:54.410
We've done a two
column proof, and we

00:10:54.410 --> 00:10:57.220
have proven that this line
segment right over here

00:10:57.220 --> 00:10:59.740
is perpendicular to that line
segment right over there.

00:10:59.740 --> 00:11:02.540
We did it with SSS,
with the SSS postulate,

00:11:02.540 --> 00:11:05.590
and the side angle
side postulate.