Measure of circumscribed angle

Angle A is a circumscribed
angle on circle O.

So this is angle
A right over here.

Then when they say it's
a circumscribed angle,

that means that the
two sides of the angle

are tangent to the circle.

So AC is tangent to
the circle at point

C. AB is tangent to
the circle at point B.

What is the measure of angle A?

Now, I encourage you
to pause the video now

and to try this out on your own.

And I'll give you a hint.

It will leverage the fact that
this is a circumscribed angle

as you could imagine.

So I'm assuming you've
given a go at it.

So the other piece
of information

they give us is that angle D,
which is an inscribed angle,

is 48 degrees and it intercepts
the same arc-- so this

is the arc that it intercepts,
arc CB I guess you could call

it-- it intercepts this
arc right over here.

It's the inscribed angle.

The central angle that
intersects that same arc

is going to be twice
the inscribed angle.

So this is going
to be 96 degrees.

I could put three markers here
just because we've already

used the double marker.

Notice, they both intercept
arc CB so some people would

say the measure of
arc CB is 96 degrees,

the central angle is 96
degrees, the inscribed angle

is going to be half
of that, 48 degrees.

So how does this help us?

Well, a key clue is that angle
is a circumscribed angle.

So that means AC and AB are
each tangent to the circle.

Well, a line that is
tangent to the circle

is going to be perpendicular to
the radius of the circle that

intersects the circle
at the same point.

So this right over here is
going to be a 90-degree angle,

and this right over here is
going to be a 90-degree angle.

OC is perpendicular to CA.

OB, which is a radius,
is perpendicular to BA,

which is a tangent line, and
they both intersect right

over here at B. Now, this
might jump out at you.

We have a quadrilateral
going on here.

ABOC is a quadrilateral,
so its sides

are going to add
up to 360 degrees.

So we could know, we
could write it this way.

We could write the
measure of angle A

plus 90 degrees plus another
90 degrees plus 96 degrees

is going to be equal
to 360 degrees.

Or another way of thinking
about it, if we subtract 180

from both sides, if we
subtract that from both sides,

we get the measure of
angle A plus 96 degrees

is going to be equal
to 180 degrees.

Or another way of
thinking about it

is the measure of angle A
or that angle A and angle O

right over here-- you
could call it angle COB--

that these are going to be
supplementary angles if they

add up to 180 degrees.

So if we subtract 96
degrees from both sides,

we get the measure
of angle A is equal

to-- I don't want to make that
look like a less than symbol,

let make it-- measure of
angle-- this one actually

looks more like a--
measure of angle A

is equal to 180 minus 96.

Let's see, 180 minus
90 would be 90,

and then we subtract another
6 gets us to 84 degrees.