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Let's say we have a circle,
and then we have a

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diameter of the circle.

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Let me draw my best diameter.

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That's pretty good.

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This right here is the diameter
of the circle or it's a

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diameter of the circle.

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That's a diameter.

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Let's say I have a triangle
where the diameter is one side

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of the triangle, and the angle
opposite that side, it's

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vertex, sits some place
on the circumference.

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So, let's say, the angle or the
angle opposite of this diameter

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sits on that circumference.

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So the triangle
looks like this.

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The triangle looks like that.

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What I'm going to show you
in this video is that

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this triangle is going
to be a right triangle.

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The 90 degree side is going
to be the side that is

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opposite this diameter.

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I don't want to label it
just yet because that would

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ruin the fun of the proof.

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Now let's see what we
can do to show this.

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Well, we have in our tool kit
the notion of an inscribed

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angle, it's relation to
a central angle that

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subtends the same arc.

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So let's look at that.

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So let's say that this is an
inscribed angle right here.

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Let's call this theta.

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Now let's say that
that's the center of

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my circle right there.

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Then this angle right here
would be a central angle.

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Let me draw another triangle
right here, another

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line right there.

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This is a central
angle right here.

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This is a radius.

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This is the same radius
-- actually this

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distance is the same.

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But we've learned several
videos ago that look, this

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angle, this inscribed angle,
it subtends this arc up here.

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The central angle that subtends
that same arc is going

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to be twice this angle.

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We proved that
several videos ago.

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So this is going to be 2theta.

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It's the central angle
subtending the same arc.

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Now, this triangle right here,
this one right here, this

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is an isosceles triangle.

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I could rotate it and
draw it like this.

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If I flipped it over it would
look like that, that, and then

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the green side would
be down like that.

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And both of these sides
are of length r.

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This top angle is 2theta.

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So all I did is I took it
and I rotated it around to

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draw it for you this way.

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This side is that
side right there.

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Since its two sides are equal,
this is isosceles, so these to

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base angles must be the same.

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That and that must be the same,
or if I were to draw it up

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here, that and that must be
the exact same base angle.

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Now let me see, I already
used theta, maybe I'll

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use x for these angles.

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So this has to be x,
and that has to be x.

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So what is x going
to be equal to?

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Well, x plus x plus 2theta
have to equal 180 degrees.

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They're all in the
same triangle.

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So let me write that down.

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We get x plus x plus 2theta,
all have to be equal to 180

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degrees, or we get 2x plus
2theta is equal to 180 degrees,

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or we get 2x is equal
to 180 minus 2theta.

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Divide both sides by 2, you get
x is equal to 90 minus theta.

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So x is equal to
90 minus theta.

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Now let's see what else
we could do with this.

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Well we could look at this
triangle right here.

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This triangle, this side over
here also has this distance

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right here is also a
radius of the circle.

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This distance over here we've
already labeled it, is

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a radius of a circle.

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So once again, this is also
an isosceles triangle.

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These two sides are equal,
so these two base angles

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have to be equal.

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So if this is theta,
this is also going to

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be equal to theta.

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And actually, we use that
information, we use to actually

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show that first result about
inscribed angles and the

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relation between them and
central angles subtending

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the same arc.

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So if this is theta, that's
theta because this is

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an isosceles triangle.

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So what is this whole
angle over here?

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Well it's going to be theta
plus 90 minus theta.

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That angle right there's
going to be theta

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plus 90 minus theta.

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Well, the thetas cancel out.

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So no matter what, as long as
one side of my triangle is the

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diameter, and then the angle or
the vertex of the angle

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opposite sits opposite of
that side, sits on the

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circumference, then this angle
right here is going to be a

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right angle, and this is going
to be a right triangle.

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So if I just were to draw
something random like this --

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if I were to just take a point
right there, like that, and

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draw it just like that,
this is a right angle.

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If I were to draw something
like that and go out like

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that, this is a right angle.

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For any of these I could
do this exact same proof.

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And in fact, the way I drew it
right here, I kept it very

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general so it would apply
to any of these triangles.

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