An inscribed angle is half of a central angle that subtends the same arc

0:01 - 0:03

What I want to do in this video
is to prove one of the more
0:03 - 0:09

useful results in geometry, and
that's that an inscribed angle
0:09 - 0:15

is just an angle whose vertex
sits on the circumference
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of the circle.
0:17 - 0:20

So that is our inscribed angle.
0:20 - 0:25

I'll denote it by psi -- I'll
use the psi for inscribed angle
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and angles in this video.
0:27 - 0:34

That psi, the inscribed angle,
is going to be exactly 1/2 of
0:34 - 0:38

the central angle that
subtends the same arc.
0:38 - 0:41

So I just used a lot a fancy
words, but I think you'll
0:41 - 0:42

get what I'm saying.
0:42 - 0:43

So this is psi.
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It is an inscribed angle.
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It sits, its vertex sits
on the circumference.
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And if you draw out the two rays
that come out from this angle
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or the two cords that define
this angle, it intersects the
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circle at the other end.
0:57 - 1:00

And if you look at the part of
the circumference of the circle
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that's inside of it, that
is the arc that is
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subtended by psi.
1:06 - 1:09

It's all very fancy words,
but I think the idea is
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pretty straightforward.
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This right here is the arc
subtended by psi, where psi is
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that inscribed angle right over
there, the vertex sitting
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on the circumference.
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Now, a central angle is an
angle where the vertex is
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sitting at the center
of the circle.
1:39 - 1:42

So let's say that this right
here -- I'll try to eyeball
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it -- that right there is
the center of the circle.
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So let me draw a central angle
that subtends this same arc.
1:51 - 1:58

So that looks like a central
angle subtending that same arc.
1:58 - 1:59

Just like that.
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Let's call this theta.
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So this angle is psi, this
angle right here is theta.
2:06 - 2:10

What I'm going to prove in this
video is that psi is always
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going to be equal
to 1/2 of theta.
2:14 - 2:18

So if I were to tell you that
psi is equal to, I don't know,
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25 degrees, then you would
immediately know that theta
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must be equal to 50 degrees.
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Or if I told you that theta was
80 degrees, then you would
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immediately know that
psi was 40 degrees.
2:29 - 2:32

So let's actually proved this.
2:32 - 2:35

So let me clear this.
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So a good place to start,
or the place I'm going to
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start, is a special case.
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I'm going to draw an inscribed
angle, but one of the chords
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that define it is going to be
the diameter of the circle.
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So this isn't going to be the
general case, this is going
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to be a special case.
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So let me see, this is the
center right here of my circle.
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I'm trying to eyeball it.
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Center looks like that.
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So let me draw a diameter.
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So the diameter
looks like that.
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Then let me define
my inscribed angle.
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This diameter is
one side of it.
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And then the other side
maybe is just like that.
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So let me call this
right here psi.
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If that's psi, this length right
here is a radius -- that's
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our radius of our circle.
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Then this length right here is
also going to be the radius of
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our circle going from the
center to the circumference.
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Your circumference is defined
by all of the points that are
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exactly a radius away
from the center.
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So that's also a radius.
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Now, this triangle right here
is an isosceles triangle.
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It has two sides
that are equal.
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Two sides that are
definitely equal.
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We know that when we have two
sides being equal, their
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base angles are also equal.
3:57 - 4:01

So this will also
be equal to psi.
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You might not recognize
it because it's
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tilted up like that.
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But I think many of us when we
see a triangle that looks like
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this, if I told you this is r
and that is r, that these two
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sides are equal, and if this is
psi, then you would also
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know that this angle is
also going to be psi.
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Base angles are equivalent
on an isosceles triangle.
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So this is psi, that is also psi.
4:27 - 4:30

Now, let me look at
the central angle.
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This is the central angle
subtending the same arc.
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Let's highlight the arc that
they're both subtending.
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This right here is the arc that
they're both going to subtend.
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So this is my central
angle right there, theta.
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Now if this angle is theta,
what's this angle going to be?
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This angle right here.
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Well, this angle is
supplementary to theta,
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so it's 180 minus theta.
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When you add these two angles
together you go 180 degrees
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around or they kind
of form a line.
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They're supplementary
to each other.
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Now we also know that these
three angles are sitting
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inside of the same triangle.
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So they must add up
to 180 degrees.
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So we get psi -- this psi plus
that psi plus psi plus this
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angle, which is 180 minus
theta plus 180 minus theta.
5:25 - 5:29

These three angles must
add up to 180 degrees.
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They're the three
angles of a triangle.
5:32 - 5:35

Now we could subtract
180 from both sides.
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psi plus psi is 2 psi minus
theta is equal to 0.
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Add theta to both sides.
5:45 - 5:49

You get 2 psi is equal to theta.
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Multiply both sides by 1/2
or divide both sides by 2.
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You get psi is equal
to 1/2 of theta.
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So we just proved what we set
out to prove for the special
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case where our inscribed angle
is defined, where one of the
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rays, if you want to view these
lines as rays, where one of the
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rays that defines this
inscribed angle is
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along the diameter.
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The diameter forms
part of that ray.
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So this is a special
case where one edge is
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sitting on the diameter.
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So already we could
generalize this.
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So now that we know that if
this is 50 that this is
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going to be 100 degrees
and likewise, right?
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Whatever psi is or whatever
theta is, psi's going to be 1/2
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of that, or whatever psi is,
theta is going to
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be 2 times that.
6:42 - 6:44

And now this will
apply for any time.
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We could use this notion any
time that -- so just using that
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result we just got, we can now
generalize it a little bit,
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although this won't apply
to all inscribed angles.
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Let's have an inscribed
angle that looks like this.
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So this situation, the center,
you can kind of view it as
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it's inside of the angle.
7:15 - 7:17

That's my inscribed angle.
7:17 - 7:19

And I want to find a
relationship between this
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inscribed angle and the central
angle that's subtending
7:22 - 7:24

to same arc.
7:24 - 7:30

So that's my central angle
subtending the same arc.
7:30 - 7:34

Well, you might say, hey, gee,
none of these ends or these
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chords that define this angle,
neither of these are diameters,
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but what we can do is
we can draw a diameter.
7:40 - 7:43

If the center is within
these two chords we
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can draw a diameter.
7:46 - 7:49

We can draw a diameter
just like that.
7:49 - 7:52

If we draw a diameter just like
that, if we define this angle
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as psi 1, that angle as psi 2.
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Clearly psi is the sum
of those two angles.
7:58 - 8:04

And we call this angle theta
1, and this angle theta 2.
8:04 - 8:07

We immediately you know that,
just using the result I just
8:07 - 8:13

got, since we have one side of
our angles in both cases being
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a diameter now, we know
that psi 1 is going to be
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equal to 1/2 theta 1.
8:22 - 8:25

And we know that psi 2 is
going to be 1/2 theta 2.
8:25 - 8:30

Psi 2 is going to
be 1/2 theta 2.
8:30 - 8:40

So psi, which is psi 1 plus psi 2,
so psi 1 plus psi 2 is going to
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be equal to these two things.
8:41 - 8:48

1/2 theta 1 plus 1/2 theta 2.
8:48 - 8:51

psi 1 plus psi 2, this is equal
to the first inscribed
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angle that we want to deal
with, just regular psi.
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That's psi.
8:55 - 8:58

And this right here, this
is equal to 1/2 times
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theta 1 plus theta 2.
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What's theta 1 plus theta 2?
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Well that's just our
original theta that
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we were dealing with.
9:08 - 9:12

So now we see that psi
is equal to 1/2 theta.
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So now we've proved it for a
slightly more general case
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where our center is inside
of the two rays that
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define that angle.
9:22 - 9:27

Now, we still haven't addressed
a slightly harder situation or
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a more general situation where
if this is the center of our
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circle and I have an inscribed
angle where the center isn't
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sitting inside of
the two chords.
9:41 - 9:42

Let me draw that.
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So that's going to be my
vertex, and I'll switch colors,
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so let's say that is one of the
chords that defines the
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angle, just like that.
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And let's say that is the
other chord that defines
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the angle just like that.
9:59 - 10:02

So how do we find the
relationship between, let's
10:02 - 10:08

call, this angle right
here, let's call it psi 1.
10:08 - 10:13

How do we find the relationship
between psi 1 and the central
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angle that subtends
this same arc?
10:16 - 10:20

So when I talk about the same
arc, that's that right there.
10:20 - 10:23

So the central angle that
subtends the same arc
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will look like this.
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Let's call that theta 1.
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What we can do is use what we
just learned when one side of
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our inscribed angle
is a diameter.
10:39 - 10:41

So let's construct that.
10:41 - 10:44

So let me draw a diameter here.
10:44 - 10:47

The result we want still is
that this should be 1/2 of
10:47 - 10:48

this, but let's prove it.
10:48 - 10:58

Let's draw a diameter
just like that.
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Let me call this angle right
here, let me call that psi 2.
11:09 - 11:15

And it is subtending this arc
right there -- let me do
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that in a darker color.
11:16 - 11:20

It is subtending this
arc right there.
11:20 - 11:22

So the central angle that
subtends that same arc,
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let me call that theta 2.
11:25 - 11:31

Now, we know from the earlier
part of this video that psi
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2 is going to be equal
to 1/2 theta 2, right?
11:38 - 11:41

They share -- the
diameter is right there.
11:41 - 11:44

The diameter is one of the
chords that forms the angle.
11:44 - 11:48

So psi 2 is going to be
equal to 1/2 theta 2.
11:50 - 11:53

This is exactly what we've been
doing in the last video, right?
11:53 - 11:55

This is an inscribed angle.
11:55 - 12:00

One of the chords that define
is sitting on the diameter.
12:00 - 12:03

So this is going to be 1/2 of
this angle, of the central
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angle that subtends
the same arc.
12:06 - 12:09

Now, let's look at
this larger angle.
12:09 - 12:12

This larger angle right here.
12:12 - 12:14

Psi 1 plus psi 2.
12:14 - 12:23

Right, that larger angle
is psi 1 plus psi 2.
12:23 - 12:29

Once again, this subtends this
entire arc right here, and it
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has a diameter as one of the
chords that defines
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this huge angle.
12:34 - 12:37

So this is going to be 1/2
of the central angle that
12:37 - 12:39

subtends the same arc.
12:39 - 12:42

We're just using what we've
already shown in this video.
12:42 - 12:47

So this is going to be equal to
1/2 of this huge central angle
12:47 - 12:51

of theta 1 plus theta 2.
12:54 - 12:57

So far we've just used
everything that we've learned
12:57 - 12:58

earlier in this video.
12:58 - 13:03

Now, we already know that psi
2 is equal to 1/2 theta 2.
13:03 - 13:06

So let me make that
substitution.
13:06 - 13:07

This is equal to that.
13:07 - 13:15

So we can say that si 1 plus
-- instead of si 2 I'll write
13:15 - 13:27

1/2 theta 2 is equal to 1/2
theta 1 plus 1/2 theta 2.
13:30 - 13:34

We can subtract 1/2 theta
2 from both sides, and
13:34 - 13:36

we get our result.
13:36 - 13:41

Si 1 is equal to 1/2 theta one.
13:41 - 13:42

And now we're done.
13:42 - 13:45

We have proven the situation
that the inscribed angle is
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always 1/2 of the central angle
that subtends the same arc,
13:51 - 13:54

regardless of whether the
center of the circle is inside
13:54 - 13:59

of the angle, outside of the
angle, whether we have a
13:59 - 14:01

diameter on one side.
14:01 - 14:06

So any other angle can be
constructed as a sum of
14:06 - 14:08

any or all of these that
we've already done.
14:08 - 14:10

So hopefully you found this
useful and now we can actually
14:10 - 14:15

build on this result to do some
more interesting
14:15 - 14:16

geometry proofs.

Title:: An inscribed angle is half of a central angle that subtends the same arc
Description:: Showing that an inscribed angle is half of a central angle that subtends the same arc

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Video Language:: English
Duration:: 14:16

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ddc419

An inscribed angle is half of a central angle that subtends the same arc

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