WEBVTT

00:00:00.690 --> 00:00:03.450
What I want to do in this video
is to prove one of the more

00:00:03.450 --> 00:00:08.980
useful results in geometry, and
that's that an inscribed angle

00:00:08.980 --> 00:00:14.950
is just an angle whose vertex
sits on the circumference

00:00:14.950 --> 00:00:17.080
of the circle.

00:00:17.080 --> 00:00:19.800
So that is our inscribed angle.

00:00:19.800 --> 00:00:24.950
I'll denote it by psi -- I'll
use the psi for inscribed angle

00:00:24.950 --> 00:00:27.170
and angles in this video.

00:00:27.170 --> 00:00:33.530
That psi, the inscribed angle,
is going to be exactly 1/2 of

00:00:33.530 --> 00:00:37.880
the central angle that
subtends the same arc.

00:00:37.880 --> 00:00:40.730
So I just used a lot a fancy
words, but I think you'll

00:00:40.730 --> 00:00:41.650
get what I'm saying.

00:00:41.650 --> 00:00:42.820
So this is psi.

00:00:42.820 --> 00:00:44.470
It is an inscribed angle.

00:00:44.470 --> 00:00:48.710
It sits, its vertex sits
on the circumference.

00:00:48.710 --> 00:00:52.570
And if you draw out the two rays
that come out from this angle

00:00:52.570 --> 00:00:56.040
or the two cords that define
this angle, it intersects the

00:00:56.040 --> 00:00:57.340
circle at the other end.

00:00:57.340 --> 00:01:00.390
And if you look at the part of
the circumference of the circle

00:01:00.390 --> 00:01:03.730
that's inside of it, that
is the arc that is

00:01:03.730 --> 00:01:06.160
subtended by psi.

00:01:06.160 --> 00:01:09.010
It's all very fancy words,
but I think the idea is

00:01:09.010 --> 00:01:09.920
pretty straightforward.

00:01:09.920 --> 00:01:28.485
This right here is the arc
subtended by psi, where psi is

00:01:28.485 --> 00:01:31.560
that inscribed angle right over
there, the vertex sitting

00:01:31.560 --> 00:01:32.400
on the circumference.

00:01:32.400 --> 00:01:37.920
Now, a central angle is an
angle where the vertex is

00:01:37.920 --> 00:01:39.460
sitting at the center
of the circle.

00:01:39.460 --> 00:01:41.880
So let's say that this right
here -- I'll try to eyeball

00:01:41.880 --> 00:01:45.510
it -- that right there is
the center of the circle.

00:01:45.510 --> 00:01:51.360
So let me draw a central angle
that subtends this same arc.

00:01:51.360 --> 00:01:58.470
So that looks like a central
angle subtending that same arc.

00:01:58.470 --> 00:01:59.390
Just like that.

00:01:59.390 --> 00:02:01.440
Let's call this theta.

00:02:01.440 --> 00:02:06.030
So this angle is psi, this
angle right here is theta.

00:02:06.030 --> 00:02:10.120
What I'm going to prove in this
video is that psi is always

00:02:10.120 --> 00:02:14.050
going to be equal
to 1/2 of theta.

00:02:14.050 --> 00:02:18.220
So if I were to tell you that
psi is equal to, I don't know,

00:02:18.220 --> 00:02:21.330
25 degrees, then you would
immediately know that theta

00:02:21.330 --> 00:02:23.090
must be equal to 50 degrees.

00:02:23.090 --> 00:02:26.080
Or if I told you that theta was
80 degrees, then you would

00:02:26.080 --> 00:02:29.300
immediately know that
psi was 40 degrees.

00:02:29.300 --> 00:02:31.500
So let's actually proved this.

00:02:31.500 --> 00:02:34.520
So let me clear this.

00:02:34.520 --> 00:02:37.730
So a good place to start,
or the place I'm going to

00:02:37.730 --> 00:02:40.460
start, is a special case.

00:02:40.460 --> 00:02:45.250
I'm going to draw an inscribed
angle, but one of the chords

00:02:45.250 --> 00:02:47.910
that define it is going to be
the diameter of the circle.

00:02:47.910 --> 00:02:50.526
So this isn't going to be the
general case, this is going

00:02:50.526 --> 00:02:51.320
to be a special case.

00:02:51.320 --> 00:02:55.325
So let me see, this is the
center right here of my circle.

00:02:55.325 --> 00:02:59.030
I'm trying to eyeball it.

00:02:59.030 --> 00:03:00.770
Center looks like that.

00:03:00.770 --> 00:03:04.210
So let me draw a diameter.

00:03:04.210 --> 00:03:06.440
So the diameter
looks like that.

00:03:06.440 --> 00:03:09.410
Then let me define
my inscribed angle.

00:03:09.410 --> 00:03:11.860
This diameter is
one side of it.

00:03:11.860 --> 00:03:15.910
And then the other side
maybe is just like that.

00:03:15.910 --> 00:03:20.520
So let me call this
right here psi.

00:03:20.520 --> 00:03:27.120
If that's psi, this length right
here is a radius -- that's

00:03:27.120 --> 00:03:29.330
our radius of our circle.

00:03:29.330 --> 00:03:33.080
Then this length right here is
also going to be the radius of

00:03:33.080 --> 00:03:35.760
our circle going from the
center to the circumference.

00:03:35.760 --> 00:03:38.130
Your circumference is defined
by all of the points that are

00:03:38.130 --> 00:03:40.340
exactly a radius away
from the center.

00:03:40.340 --> 00:03:43.610
So that's also a radius.

00:03:43.610 --> 00:03:47.920
Now, this triangle right here
is an isosceles triangle.

00:03:47.920 --> 00:03:49.890
It has two sides
that are equal.

00:03:49.890 --> 00:03:51.880
Two sides that are
definitely equal.

00:03:51.880 --> 00:03:54.630
We know that when we have two
sides being equal, their

00:03:54.630 --> 00:03:57.290
base angles are also equal.

00:03:57.290 --> 00:04:00.640
So this will also
be equal to psi.

00:04:00.640 --> 00:04:02.130
You might not recognize
it because it's

00:04:02.130 --> 00:04:03.180
tilted up like that.

00:04:03.180 --> 00:04:05.720
But I think many of us when we
see a triangle that looks like

00:04:05.720 --> 00:04:10.940
this, if I told you this is r
and that is r, that these two

00:04:10.940 --> 00:04:17.860
sides are equal, and if this is
psi, then you would also

00:04:17.860 --> 00:04:20.830
know that this angle is
also going to be psi.

00:04:20.830 --> 00:04:23.930
Base angles are equivalent
on an isosceles triangle.

00:04:23.930 --> 00:04:26.720
So this is psi, that is also psi.

00:04:26.720 --> 00:04:29.770
Now, let me look at
the central angle.

00:04:29.770 --> 00:04:32.710
This is the central angle
subtending the same arc.

00:04:32.710 --> 00:04:35.920
Let's highlight the arc that
they're both subtending.

00:04:35.920 --> 00:04:40.300
This right here is the arc that
they're both going to subtend.

00:04:40.300 --> 00:04:44.350
So this is my central
angle right there, theta.

00:04:44.350 --> 00:04:49.000
Now if this angle is theta,
what's this angle going to be?

00:04:49.000 --> 00:04:50.620
This angle right here.

00:04:50.620 --> 00:04:53.010
Well, this angle is
supplementary to theta,

00:04:53.010 --> 00:04:56.640
so it's 180 minus theta.

00:04:56.640 --> 00:04:59.560
When you add these two angles
together you go 180 degrees

00:04:59.560 --> 00:05:01.750
around or they kind
of form a line.

00:05:01.750 --> 00:05:03.790
They're supplementary
to each other.

00:05:03.790 --> 00:05:06.740
Now we also know that these
three angles are sitting

00:05:06.740 --> 00:05:08.260
inside of the same triangle.

00:05:08.260 --> 00:05:12.030
So they must add up
to 180 degrees.

00:05:12.030 --> 00:05:19.300
So we get psi -- this psi plus
that psi plus psi plus this

00:05:19.300 --> 00:05:25.420
angle, which is 180 minus
theta plus 180 minus theta.

00:05:25.420 --> 00:05:29.130
These three angles must
add up to 180 degrees.

00:05:29.130 --> 00:05:31.740
They're the three
angles of a triangle.

00:05:31.740 --> 00:05:34.605
Now we could subtract
180 from both sides.

00:05:37.140 --> 00:05:43.260
psi plus psi is 2 psi minus
theta is equal to 0.

00:05:43.260 --> 00:05:44.840
Add theta to both sides.

00:05:44.840 --> 00:05:48.770
You get 2 psi is equal to theta.

00:05:48.770 --> 00:05:52.850
Multiply both sides by 1/2
or divide both sides by 2.

00:05:52.850 --> 00:05:56.680
You get psi is equal
to 1/2 of theta.

00:05:56.680 --> 00:06:00.070
So we just proved what we set
out to prove for the special

00:06:00.070 --> 00:06:07.120
case where our inscribed angle
is defined, where one of the

00:06:07.120 --> 00:06:11.200
rays, if you want to view these
lines as rays, where one of the

00:06:11.200 --> 00:06:15.220
rays that defines this
inscribed angle is

00:06:15.220 --> 00:06:17.180
along the diameter.

00:06:17.180 --> 00:06:19.200
The diameter forms
part of that ray.

00:06:19.200 --> 00:06:21.720
So this is a special
case where one edge is

00:06:21.720 --> 00:06:23.760
sitting on the diameter.

00:06:23.760 --> 00:06:27.660
So already we could
generalize this.

00:06:27.660 --> 00:06:30.580
So now that we know that if
this is 50 that this is

00:06:30.580 --> 00:06:32.820
going to be 100 degrees
and likewise, right?

00:06:32.820 --> 00:06:37.460
Whatever psi is or whatever
theta is, psi's going to be 1/2

00:06:37.460 --> 00:06:40.450
of that, or whatever psi is,
theta is going to

00:06:40.450 --> 00:06:41.830
be 2 times that.

00:06:41.830 --> 00:06:44.110
And now this will
apply for any time.

00:06:44.110 --> 00:06:55.440
We could use this notion any
time that -- so just using that

00:06:55.440 --> 00:06:59.460
result we just got, we can now
generalize it a little bit,

00:06:59.460 --> 00:07:02.890
although this won't apply
to all inscribed angles.

00:07:02.890 --> 00:07:05.090
Let's have an inscribed
angle that looks like this.

00:07:10.680 --> 00:07:12.980
So this situation, the center,
you can kind of view it as

00:07:12.980 --> 00:07:15.470
it's inside of the angle.

00:07:15.470 --> 00:07:17.150
That's my inscribed angle.

00:07:17.150 --> 00:07:18.890
And I want to find a
relationship between this

00:07:18.890 --> 00:07:22.450
inscribed angle and the central
angle that's subtending

00:07:22.450 --> 00:07:24.360
to same arc.

00:07:24.360 --> 00:07:29.880
So that's my central angle
subtending the same arc.

00:07:29.880 --> 00:07:33.550
Well, you might say, hey, gee,
none of these ends or these

00:07:33.550 --> 00:07:37.310
chords that define this angle,
neither of these are diameters,

00:07:37.310 --> 00:07:40.400
but what we can do is
we can draw a diameter.

00:07:40.400 --> 00:07:43.300
If the center is within
these two chords we

00:07:43.300 --> 00:07:46.100
can draw a diameter.

00:07:46.100 --> 00:07:48.920
We can draw a diameter
just like that.

00:07:48.920 --> 00:07:51.680
If we draw a diameter just like
that, if we define this angle

00:07:51.680 --> 00:07:55.430
as psi 1, that angle as psi 2.

00:07:55.430 --> 00:07:58.320
Clearly psi is the sum
of those two angles.

00:07:58.320 --> 00:08:04.350
And we call this angle theta
1, and this angle theta 2.

00:08:04.350 --> 00:08:07.240
We immediately you know that,
just using the result I just

00:08:07.240 --> 00:08:12.540
got, since we have one side of
our angles in both cases being

00:08:12.540 --> 00:08:18.260
a diameter now, we know
that psi 1 is going to be

00:08:18.260 --> 00:08:22.010
equal to 1/2 theta 1.

00:08:22.010 --> 00:08:24.870
And we know that psi 2 is
going to be 1/2 theta 2.

00:08:24.870 --> 00:08:30.140
Psi 2 is going to
be 1/2 theta 2.

00:08:30.140 --> 00:08:39.850
So psi, which is psi 1 plus psi 2,
so psi 1 plus psi 2 is going to

00:08:39.850 --> 00:08:41.120
be equal to these two things.

00:08:41.120 --> 00:08:47.580
1/2 theta 1 plus 1/2 theta 2.

00:08:47.580 --> 00:08:51.180
psi 1 plus psi 2, this is equal
to the first inscribed

00:08:51.180 --> 00:08:53.850
angle that we want to deal
with, just regular psi.

00:08:53.850 --> 00:08:54.980
That's psi.

00:08:54.980 --> 00:08:58.350
And this right here, this
is equal to 1/2 times

00:08:58.350 --> 00:09:00.960
theta 1 plus theta 2.

00:09:00.960 --> 00:09:03.960
What's theta 1 plus theta 2?

00:09:03.960 --> 00:09:06.470
Well that's just our
original theta that

00:09:06.470 --> 00:09:08.490
we were dealing with.

00:09:08.490 --> 00:09:12.080
So now we see that psi
is equal to 1/2 theta.

00:09:12.080 --> 00:09:14.710
So now we've proved it for a
slightly more general case

00:09:14.710 --> 00:09:20.020
where our center is inside
of the two rays that

00:09:20.020 --> 00:09:21.640
define that angle.

00:09:21.640 --> 00:09:27.100
Now, we still haven't addressed
a slightly harder situation or

00:09:27.100 --> 00:09:33.660
a more general situation where
if this is the center of our

00:09:33.660 --> 00:09:39.420
circle and I have an inscribed
angle where the center isn't

00:09:39.420 --> 00:09:40.990
sitting inside of
the two chords.

00:09:40.990 --> 00:09:41.820
Let me draw that.

00:09:41.820 --> 00:09:48.800
So that's going to be my
vertex, and I'll switch colors,

00:09:48.800 --> 00:09:51.540
so let's say that is one of the
chords that defines the

00:09:51.540 --> 00:09:53.320
angle, just like that.

00:09:53.320 --> 00:09:57.860
And let's say that is the
other chord that defines

00:09:57.860 --> 00:09:59.170
the angle just like that.

00:09:59.170 --> 00:10:02.500
So how do we find the
relationship between, let's

00:10:02.500 --> 00:10:07.910
call, this angle right
here, let's call it psi 1.

00:10:07.910 --> 00:10:13.050
How do we find the relationship
between psi 1 and the central

00:10:13.050 --> 00:10:16.160
angle that subtends
this same arc?

00:10:16.160 --> 00:10:19.530
So when I talk about the same
arc, that's that right there.

00:10:19.530 --> 00:10:22.720
So the central angle that
subtends the same arc

00:10:22.720 --> 00:10:23.660
will look like this.

00:10:28.150 --> 00:10:32.910
Let's call that theta 1.

00:10:32.910 --> 00:10:36.770
What we can do is use what we
just learned when one side of

00:10:36.770 --> 00:10:39.350
our inscribed angle
is a diameter.

00:10:39.350 --> 00:10:41.135
So let's construct that.

00:10:41.135 --> 00:10:44.260
So let me draw a diameter here.

00:10:44.260 --> 00:10:47.010
The result we want still is
that this should be 1/2 of

00:10:47.010 --> 00:10:48.180
this, but let's prove it.

00:10:48.180 --> 00:10:57.560
Let's draw a diameter
just like that.

00:10:57.560 --> 00:11:09.490
Let me call this angle right
here, let me call that psi 2.

00:11:09.490 --> 00:11:14.770
And it is subtending this arc
right there -- let me do

00:11:14.770 --> 00:11:16.140
that in a darker color.

00:11:16.140 --> 00:11:19.770
It is subtending this
arc right there.

00:11:19.770 --> 00:11:22.360
So the central angle that
subtends that same arc,

00:11:22.360 --> 00:11:25.300
let me call that theta 2.

00:11:25.300 --> 00:11:30.890
Now, we know from the earlier
part of this video that psi

00:11:30.890 --> 00:11:37.600
2 is going to be equal
to 1/2 theta 2, right?

00:11:37.600 --> 00:11:40.760
They share -- the
diameter is right there.

00:11:40.760 --> 00:11:44.300
The diameter is one of the
chords that forms the angle.

00:11:44.300 --> 00:11:47.500
So psi 2 is going to be
equal to 1/2 theta 2.

00:11:50.140 --> 00:11:52.810
This is exactly what we've been
doing in the last video, right?

00:11:52.810 --> 00:11:55.430
This is an inscribed angle.

00:11:55.430 --> 00:11:59.550
One of the chords that define
is sitting on the diameter.

00:11:59.550 --> 00:12:02.740
So this is going to be 1/2 of
this angle, of the central

00:12:02.740 --> 00:12:05.980
angle that subtends
the same arc.

00:12:05.980 --> 00:12:09.000
Now, let's look at
this larger angle.

00:12:09.000 --> 00:12:11.680
This larger angle right here.

00:12:11.680 --> 00:12:14.240
Psi 1 plus psi 2.

00:12:14.240 --> 00:12:22.720
Right, that larger angle
is psi 1 plus psi 2.

00:12:22.720 --> 00:12:28.680
Once again, this subtends this
entire arc right here, and it

00:12:28.680 --> 00:12:32.100
has a diameter as one of the
chords that defines

00:12:32.100 --> 00:12:34.310
this huge angle.

00:12:34.310 --> 00:12:37.380
So this is going to be 1/2
of the central angle that

00:12:37.380 --> 00:12:38.580
subtends the same arc.

00:12:38.580 --> 00:12:42.270
We're just using what we've
already shown in this video.

00:12:42.270 --> 00:12:47.390
So this is going to be equal to
1/2 of this huge central angle

00:12:47.390 --> 00:12:51.370
of theta 1 plus theta 2.

00:12:54.310 --> 00:12:56.530
So far we've just used
everything that we've learned

00:12:56.530 --> 00:12:58.160
earlier in this video.

00:12:58.160 --> 00:13:03.160
Now, we already know that psi
2 is equal to 1/2 theta 2.

00:13:03.160 --> 00:13:05.630
So let me make that
substitution.

00:13:05.630 --> 00:13:07.030
This is equal to that.

00:13:07.030 --> 00:13:15.330
So we can say that si 1 plus
-- instead of si 2 I'll write

00:13:15.330 --> 00:13:26.630
1/2 theta 2 is equal to 1/2
theta 1 plus 1/2 theta 2.

00:13:30.340 --> 00:13:34.020
We can subtract 1/2 theta
2 from both sides, and

00:13:34.020 --> 00:13:35.740
we get our result.

00:13:35.740 --> 00:13:40.900
Si 1 is equal to 1/2 theta one.

00:13:40.900 --> 00:13:41.970
And now we're done.

00:13:41.970 --> 00:13:44.990
We have proven the situation
that the inscribed angle is

00:13:44.990 --> 00:13:50.680
always 1/2 of the central angle
that subtends the same arc,

00:13:50.680 --> 00:13:53.980
regardless of whether the
center of the circle is inside

00:13:53.980 --> 00:13:58.990
of the angle, outside of the
angle, whether we have a

00:13:58.990 --> 00:14:00.950
diameter on one side.

00:14:00.950 --> 00:14:05.860
So any other angle can be
constructed as a sum of

00:14:05.860 --> 00:14:08.300
any or all of these that
we've already done.

00:14:08.300 --> 00:14:10.190
So hopefully you found this
useful and now we can actually

00:14:10.190 --> 00:14:14.630
build on this result to do some
more interesting

00:14:14.630 --> 00:14:16.460
geometry proofs.