### Video Transcript

In this video, we will learn how to
find the measures of inscribed angles using the relationship between angles and
arcs.

Before we talk about these angle
relationships, letβs remember what an inscribed angle is. Itβs an angle where the vertex and
two endpoints all lie on the circumference of the circle, on the outside of the
circle. We could measure this inscribed
angle in degrees. If this inscribed angle measures π
degrees, then the arc created between these two endpoints will be two π
degrees. Another way to say that is an
inscribed angle measures half of the subtended arc thatβs created by that angle. If we have another inscribed angle,
which has the same endpoints as the first one, this angle will also measure π
degrees because π is one-half of the arc thatβs created by these endpoints, which
here is two π.

Itβs also worth noting a special
case. The special case is the case where
the angle inscribed has endpoints that are at either end of a circleβs diameter. In this case, the subtended arc is
180 degrees, which makes the inscribed angle a right angle. And again, we can move that vertex
and still create a right angle as long as the end points donβt move.

Before we move on, we should also
remind ourselves of central angles. In a central angle, the vertex is
the center of the circle. And when weβre dealing with a
central angle, its angle measure will be equal to the subtended arcβs angle
measure. The way weβve drawn it here, it
will be equal to two π. And that means when an inscribed
angle shares the endpoints with a central angle, the inscribed angle will be half
the central angle.

Another thing we should know about
arcs and circles is what happens when we have parallel chords. If we have chords like these that
run parallel, arc π΄π· will be equal to arc π΅πΆ. That is to say, in a circle, arcs
between parallel chords are congruent. In the circle weβve drawn here, the
measure of arc π΄π· will be equal to the measure of arc π΅πΆ. Weβre now ready to use these circle
theorems to calculate unknown angles.

In the figure, π is the center and
the measure of angle ππ΄π΅ equals 59.5 degrees. What is the measure of angle
π΄ππ΅? What is the measure of angle
π΄πΆπ΅?

Weβre told that π is the center of
this circle and that the measure of angle ππ΄π΅ is 59.5 degrees. We want to find the measure of
angle π΄ππ΅ and the measure of π΄πΆπ΅. We see that the points π΄, π, and
π΅ form a triangle. Both line segment ππ΅ and line
segment ππ΄ are radii of this circle because any line drawn from the center of the
circle to the circumference of the circle will be a radius. This means we can say that line
segment ππ΄ is equal to line segment ππ΅. And it will mean that triangle
π΄ππ΅ is an isosceles triangle.

In an isosceles triangle, the two
angles opposite the radii are equal to each other. And that means we could say that
angle π΄π΅π is also equal to 59.5 degrees. Since these three angles form a
triangle, they must sum to 180 degrees. And so, we substitute the values we
do know for angle ππ΄π΅ and angle π΄π΅π. We add the two angles we know, and
we get 119 degrees. And then to solve for angle π΄ππ΅,
we subtract 119 degrees from both sides, and we find that angle π΄ππ΅ is equal to
61 degrees. Thatβs the answer to part one.

Part two is a little bit less
straightforward. We notice that both of these angles
share the endpoints π΄, π΅, which means theyβre both subtended by the arc π΄π΅. But we need to make a clarification
here. Angle π΄ππ΅ is a central angle
that is subtended by arc π΄π΅, while angle π΄πΆπ΅ is an inscribed angle subtended by
arc π΄π΅. And we remember that the central
angle subtended by two points on a circle is twice the inscribed angle subtended by
those two points. We might see it represented
something like this: if the central angle measures two π, the inscribed angle
subtended by the same points will be π degrees.

Based on that, we can say that the
measure of angle π΄πΆπ΅ will be equal to one-half the measure of angle π΄ππ΅. So, we plug in 61 degrees for angle
π΄ππ΅. Half of 61 degrees is 30.5
degrees. And so, the measure of angle π΄πΆπ΅
equals 30.5 degrees.

Hereβs another example.

From the figure, what is π₯?

Letβs start with what we know. We have angle π΄πΆπ΅, which
measures 101 degrees. And we have angle π΄ππ΅. In this case, weβre talking about
the reflex of angle π΄ππ΅. Thatβs the one thatβs greater than
180 degrees, which measures two π₯ plus eight degrees. Angle π΄πΆπ΅ and angle π΄ππ΅ share
the endpoints π΄ and π΅. But because the vertex of angle
π΄ππ΅ is the center of the circle, we say that angle π΄ππ΅ is a central angle for
this circle. While the vertex for angle π΄πΆπ΅
is on the outside of the circle, making angle π΄πΆπ΅ an inscribed angle of the
circle. And these three facts point us to
the central angle theorem.

And the central angle theorem tells
us that when a central angle and an inscribed angle share the same endpoints, the
central angle will be two times that of the inscribed angle. In this diagram, the inscribed
angle is π degrees, and that would make the central angle two π degrees. By this, we can say that the
measure of angle π΄ππ΅ is going to be equal to two times the measure of angle
π΄πΆπ΅. The measure of the central angle
will be equal to two times the measure of the inscribed angle. And so, we can say that two π₯ plus
eight will be equal to two times 101. When we multiply two times 101, we
get 202. And now, weβre ready to solve for
π₯. Subtract eight from both sides. Two π₯ equals 194. Then, divide both sides by two, and
we find that π₯ equals 97.

In our next example, we have some
intersecting chords in a circle.

Given that the measure of angle
π΄π΅π· equals 44 degrees and the measure of angle πΆπΈπ΄ equals 72 degrees, find π₯,
π¦, and π§.

Letβs start by listing what we
know. Angle π΄π΅π· equals 44 degrees,
angle πΆπΈπ΄ measures 72 degrees. These two chords intersect at point
πΈ. And that means we can say that
angle π΅πΈπ· and angle πΆπΈπ΄ are vertical angles, which means their measure will be
equal to one another. They are congruent angles. And in this case, that means that
angle π΅πΈπ· is also equal to 72 degrees. The points πΈ, π΅, and π· form a
triangle, which means that their three angles must sum to 180 degrees. And we can substitute what we know
for these three angles into this equation. 72 plus 44 equals 116. 116 plus π§ equals 180. So, we subtract 116 from both
sides. And we find that π§ equals 64
degrees.

We wonβt be able to follow the same
procedure to find π₯ and π¦. So, weβll need to think about some
of the circle theorems. If we look at inscribed angle π΅,
we see that it has endpoints along the circle at π΄ and π· and that its intercepted
arc is arc π΄π·. We could write them as arc π΄π·
intercepts angle π΄π΅π·. But thereβs another angle in this
circle that also intercepts the same arc, and that would be angle π΄πΆπ·. Because both of these angles are
subtended by the same arc, we can say that the measure of angle π΄πΆπ· will be equal
to the measure of angle π΄π΅π·. And that means π₯ will be equal to
44 degrees.

And because all three angles need
to sum to 180 degrees, we can tell that angle π¦ is going to be equal to 64
degrees. If we wanted to confirm this, we
could see that angle πΆπ΄π΅ intercepts arc πΆπ΅ and angle πΆπ·π΅ intercepts arc
πΆπ΅. And so, we found that π₯ equals 44
degrees, and both π¦ and π§ equals 64 degrees.

In our next example, weβll have a
diameter to consider.

Given that line segment π΄π΅ is a
diameter in circle π and the measure of angle π΅ππ· equals 59 degrees, find the
measure of angle π΄πΆπ· in degrees.

Letβs put what we know into the
diagram. Angle π΅ππ· measures 59 degrees,
and weβre trying to find the measure of angle π΄πΆπ·. If we start with what we know about
angle π΅ππ·, since π΅ππ· has a vertex at the center of the circle, π΅ππ· is a
central angle. And because angle π΅ππ· is a
central angle, its subtended arc, arc π΅π·, also measures 59 degrees. Weβre also interested in angle
π΄πΆπ·. But angle π΄πΆπ· is not a central
angle. Itβs an inscribed angle because its
vertex is on the circumference of the circle, as are both endpoints.

The arc associated with angle
π΄πΆπ· would be arc π΄π·. We have a partial measurement for
this arc, but weβre missing the distance from π΄ to π΅. But because we know that π΄π΅ is a
diameter, it cuts the circle in half. And that means the measure of arc
π΄π΅ is 180 degrees. If arc π΄π΅ equals 180 and arc π΅π·
equals 59 degrees, we can say the measure of arc π΄π· is equal to the measure of arc
π΄π΅ plus the measure of arc π΅π·.

If we plug in what we know, the
measure of arc π΄π· is 239 degrees. Because angle π΄πΆπ· is an
inscribed angle and it has a subtended arc measure of 239 degrees, we can find out
the exact measure of angle π΄πΆπ·. The measure of the inscribed angle
π΄πΆπ· will be one-half its subtended arc, arc π΄π·. Since that arc is 239 degrees, we
take half of that and we get 119.5 degrees for the measure of angle π΄πΆπ·.

In our final example, weβll look at
how parallel chords can give us information about arc measures.

Given that the line segment π΄π΅ is
a diameter of the circle and line segment π·πΆ is parallel to line segment π΄π΅,
find the measure of angle π΄πΈπ·.

Weβre interested in the measure of
angle π΄πΈπ·; thatβs this measure. And weβve been given a few other
pieces of information. We know line segment π·πΆ is
parallel to line segment π΄π΅. We know line segment π΄π΅ is the
diameter. And on the figure, angle πΆπ΅π΄ has
been labeled as 68.5 degrees.

At first, it might not seem like
thereβs a very clear direction for where to go here. But if we start with the measure of
angle πΆπ΄π΅, using that information, we could find the measure of arc πΆπ΄. Since angle πΆπ΄π΅ is an inscribed
angle, its arc will be two times the measure of that inscribed angle. Arc π΄πΆ will then be equal to two
times 68.5, which is 137 degrees. And because we know that line
segment π΄π΅ is a diameter, arc π΄π΅ must be equal to 180 degrees. We can also say that arc π΄π΅ will
be equal to arc π΅πΆ plus arc πΆπ΄.

We know π΄π΅ needs to be 180
degrees and arc πΆπ΄ is 137 degrees. To solve for the measure of arc
π΅πΆ, we can subtract 137 from both sides of the equation. And we get the measure of arc π΅πΆ
is 43 degrees. And hereβs where our parallel
chords come into play. When you have parallel chords,
their intercepted arcs are going to be congruent. And that means because arc πΆπ΅
equals 43 degrees, arc π·π΄ also equals 43 degrees.

And at this point, we began to see
that arc π·π΄ is subtended by the angle π΄πΈπ·. Since angle π΄πΈπ· is an inscribed
angle, its angle measure, the measure of angle π΄πΈπ·, is going to be equal to
one-half the measure of arc π΄π·. We know that the measure of arc
π΄π· is 43 degrees, and one-half of 43 is 21.5. And so, we can say that the measure
of angle π΄πΈπ· is 21.5 degrees.

Before we finish, letβs quickly
review the key points. If you have a central angle that
measures two π degrees, its intercepted arc will also measure two π degrees. While an inscribed angle that
intercepts the same arc will have half the angle measure, only π degrees. We could say it like this: the
central angle subtended by two points on a circle is twice the inscribed angle
subtended by those same two points. We also can say that the angles
subtended by the same arc on a circle will be equal. And finally, the arcs between
parallel chords will always be congruent.