Video Transcript
In this video, we’ll learn how to
find the measures of inscribed angles subtended by the same arc or by congruent
arcs. To begin, we’ll recap the meaning
of some of those key terms before looking at a theorem that will help us to solve
problems involving missing angles. An inscribed angle is the angle
that’s formed by the intersection of a pair of chords on the circumference of a
circle. In the diagram, angle 𝐴𝐵𝐶 is the
inscribed angle. Now, this angle is also said to be
subtended by the arc 𝐴𝐶.
There are a number of properties
that apply to such angles. And in this video, we’re mostly
going to investigate one such property. That is, angles subtended by the
same arc are equal. In this diagram, that means the
measure of angle 𝐴𝐷𝐶 is equal to the measure of angle 𝐴𝐵𝐶, since both angles
are subtended from arc 𝐴𝐶. In a similar way, the measure of
angle 𝐷𝐴𝐵 is equal to the measure of angle 𝐷𝐶𝐵. This time, they’re both subtended
from arc 𝐵𝐷. This property is sometimes
equivalently referred to as angles in the same segment are equal. But it’s also sometimes informally
referred to as the bow tie property, since the pair of inscribed angles form the
shape of a bow tie.
It’s important to note that that’s
an informal definition and should not be referred to in a mathematical proof or
otherwise. And an incredibly powerful aspect
of this property is that we can construct any number of angles subtended from arc
𝐴𝐶 and they’ll all be equal as shown. Similarly, any number of angles can
be subtended from arc 𝐵𝐷 or even 𝐵𝐸. And those will also all be
equal. So before we demonstrate the
application of this property, let’s have a look at a very short geometric proof.
To complete this proof, we’ll add
the center of the circle. Let’s define that to be 𝑂. And then we’re going to construct a
pair of radii, that is, the radii 𝐴𝑂 and 𝑂𝐶. Then we apply a known property. That is, the inscribed angle is
half the central angle that subtends the same arc. Or more simply, the angle at the
center is double the angle at the circumference. Then we’re going to define this
angle at the center, angle 𝐴𝑂𝐶, as being equal to two 𝑥.
Now we could have alternatively
chosen 𝑥 degrees or 𝑏 or 𝑦. But it does make sense to choose a
multiple of two because it will make the further calculations a little bit more
simple. Then angle 𝐴𝐷𝐶 is half the size
of this. So it’s a half times two 𝑥, which
is simply 𝑥 degrees. But then we can apply the same rule
to calculate the measure of angle 𝐴𝐵𝐶. Once again, it’s half the measure
of the angle at the center, so it’s half times two 𝑥, which is 𝑥 degrees. And so we can conclude that these
angles are congruent. The measure of angle 𝐴𝐵𝐶 is
equal to the measure of angle 𝐴𝐷𝐶, as we required. So now we have the property and a
proof; we’re going to demonstrate a simple application.
Given that the measure of angle
𝐵𝐴𝐷 is equal to 36 degrees and the measure of angle 𝐶𝐵𝐴 is equal to 37
degrees, find the measure of angle 𝐵𝐶𝐷 and the measure of angle 𝐶𝐷𝐴.
Let’s begin by adding the angles
that we know onto the diagram. The measure of angle 𝐵𝐴𝐷 is
equal to 36 degrees, and the measure of angle 𝐶𝐵𝐴 is equal to 37. We are looking to calculate the
measure of angle 𝐵𝐶𝐷, which is this one, and the measure of angle 𝐶𝐷𝐴, which
is this one. We now observe that the first
unknown angle, that’s 𝐵𝐶𝐷, is subtended by the same arc 𝐵𝐷 as angle 𝐵𝐴𝐷. And we know that inscribed angles
subtended by the same arc are equal. So the measure of angle 𝐵𝐴𝐷 must
be equal to the measure of angle 𝐵𝐶𝐷. But of course, we’ve now seen that
that’s 36 degrees.
In a similar way, we observed that
angle 𝐴𝐷𝐶 is subtended from the same arc as angle 𝐴𝐵𝐶. And so, these two angles are
congruent. The measure of angle 𝐴𝐵𝐶 must be
equal to the measure of angle 𝐴𝐷𝐶. And that’s 37. So, using the property of inscribed
angles subtended from the same arc, we find the measure of angle 𝐵𝐶𝐷 is equal to
36 degrees and the measure of angle 𝐶𝐷𝐴 is equal to 37 degrees.
Now, in this example, we looked at
how to solve problems involving numerical values for the angles. In our next example, let’s have a
look at how to apply the same property, but to solve problems involving algebraic
expressions.
If the measure of angle 𝐵𝐴𝐷 is
equal to two 𝑥 plus two degrees and the measure of angle 𝐵𝐶𝐷 is equal to 𝑥 plus
18 degrees, determine the value of 𝑥.
Let’s begin by adding the measure
of angle 𝐵𝐴𝐷 and the measure of angle 𝐵𝐶𝐷 to the diagram. In doing so, we see that each of
these inscribed angles is subtended from arc 𝐵𝐷. And so we quote one of the theorems
that we use when working with inscribed angles. That is, angles subtended by the
same arc are equal, or alternatively angles in the same segment are equal. This must mean that angle 𝐵𝐴𝐷 is
equal to angle 𝐵𝐶𝐷. This allows us to form and solve an
equation in 𝑥. The measure of angle 𝐵𝐴𝐷 is two
𝑥 plus two degrees, and the measure of angle 𝐵𝐶𝐷 is 𝑥 plus 18 degrees. So two 𝑥 plus two must be equal to
𝑥 plus 18.
To solve this equation for 𝑥,
let’s begin by subtracting 𝑥 from both sides, giving us 𝑥 plus two equals 18. Finally, we can isolate the 𝑥 by
subtracting two from both sides. 18 minus two is equal to 16. And so we’ve determined the value
of 𝑥; it’s 16.
Now, it probably comes as no
surprise that we can extend the properties of inscribed angles to work with distinct
circles or even congruent arcs. In particular, if a pair of circles
are congruent, then inscribed angles subtended by congruent arcs will be equal. What this means for our diagram is
that if arcs 𝐴𝐶 and 𝐷𝐹 are congruent, then the measure of angle 𝐴𝐵𝐶 must be
equal to the measure of angle 𝐷𝐸𝐹.
And so we see that whilst it might
be tempting to look for the typical bow tie shape to help us solve problems
involving inscribed angles, this isn’t always the most sensible route. In our next example, we’ll
demonstrate that.
Given that the measure of angle
𝐹𝐸𝐷 is equal to 14 degrees and the measure of angle 𝐶𝐵𝐴 is equal to two 𝑥
minus 96 degrees, calculate the value of 𝑥.
So let’s look at the diagram. We quickly see that arc 𝐴𝐶 is
congruent to arc 𝐷𝐹. And we know that inscribed angles
subtended by congruent arcs are going to be equal in measure. So this means that angle 𝐴𝐵𝐶 is
going to be equal in measure to angle 𝐷𝐸𝐹. We’re in fact told that the measure
of angle 𝐴𝐵𝐶 or 𝐶𝐵𝐴 is two 𝑥 minus 96. And the measure of angle 𝐹𝐸𝐷,
which is the same as 𝐷𝐸𝐹, is 14 degrees. Since these angles are equal, we
can form and solve an equation in 𝑥. That is, two 𝑥 minus 96 equals
14. To solve for 𝑥, we add 96 to both
sides, giving us two 𝑥 is equal to 110. And finally, we divide through by
two, and that gives us 𝑥 is equal to 55. And so given the information about
angles 𝐹𝐸𝐷 and 𝐶𝐵𝐴, we can deduce that 𝑥 is equal to 55.
Now, in all our previous examples,
we’ve considered congruent, that is, identical, circles. What do we do if we’re given a pair
of concentric circles? Remember, concentric circles are a
pair of circles which share the same center. We also know that any two circles
will always be similar to one another. And so, we can say that inscribed
angles subtended by two arcs of equal measure in these circles with the same center
must be equal to one another. In our next example, we’ll
demonstrate what that would look like.
In the figure, line segment 𝐴𝐸
and line segment 𝐵𝐶 pass through the midpoint of the circles. Given that the measure of angle
𝐹𝐸𝐷 is 50 degrees and the measure of angle 𝐶𝐵𝐴 is equal to two 𝑥 minus 10
degrees, find 𝑥.
We begin by adding the relevant
measurements to our diagram. 𝐹𝐸𝐷 is 50 degrees, and 𝐶𝐵𝐴 is
two 𝑥 minus 10 degrees. Now, we do know that angles
subtended from the same arc are equal. But we also know that angles
subtended from arcs with equal measure are equal. And this is really useful when
we’re working with a pair of concentric circles, because we’re able to say that the
measure of arc 𝐹𝐷 is equal to the measure of arc 𝐶𝐴. They’re both equal to this angle
here. Since the measure of those two arcs
are equal, then the measure of any angles subtended from the arcs must also be
equal. In other words, the measure of
angle 𝐹𝐸𝐷 must be equal to the measure of angle 𝐶𝐵𝐴.
And so, we can say that 50 must be
equal to two 𝑥 minus 10. Then, we simply have an equation
that we can solve for 𝑥. We’ll begin by adding 10 to both
sides, giving us 60 equals two 𝑥. And then we divide through by two,
giving us 30 is equal to 𝑥, or 𝑥 is equal to 30.
In our previous examples, we’ve
used the properties of inscribed angles in a circle to find missing values. Now, it in fact follows that we can
apply a reverse property to prove statements about circles. In other words, if we have two
congruent angles subtended from the same line segment and on the same side of that
line segment, then their vertices and the endpoints of the line segment must lie on
a circle. In the diagram, for instance,
because the measure of angle 𝐴𝐵𝐶 is equal to the measure of angle 𝐴𝐷𝐶 and
these two angles are subtended from the line segment 𝐴𝐶 and they’re on the same
side, then the points 𝐴, 𝐵, 𝐶, 𝐷 must all lie on a circle. In our final example, we’ll use
this information to decide whether a circle passes through four given points.
Given that the measure of angle
𝐵𝐶𝐴 equals 61 degrees and the measure of angle 𝐷𝐴𝐵 equals 98 degrees, can a
circle pass through the points 𝐴, 𝐵, 𝐶, and 𝐷?
Remember, if there are a pair of
congruent angles subtended by the same line segment and on the same side of it, then
their vertices and the segment’s endpoints lie on a circle in which that segment is
a chord. Well, we have a line segment 𝐵𝐴,
from which angle 𝐵𝐶𝐴 and 𝐵𝐷𝐴 are subtended. The angles lie on the same side of
that line segment. So if angle 𝐵𝐶𝐴 is equal to
angle 𝐵𝐷𝐴, then all four of our points must lie on the circumference of a
circle. Now, we’re given that the measure
of angle 𝐵𝐶𝐴 is 61 degrees and the measure of angle 𝐵𝐴𝐷 is 98.
Since triangle 𝐵𝐷𝐴 is isosceles,
we can calculate the measure of angle 𝐵𝐷𝐴 by subtracting 98 from 180 and then
dividing by two. And that gives us that the measure
of angle 𝐵𝐷𝐴 is 41 degrees. So we see that the measure of angle
𝐵𝐶𝐴 is not equal to the measure of angle 𝐵𝐷𝐴. Since these angles are not equal,
we observe that a circle cannot pass through the points, and the answer is no.
Let’s recap the key concepts from
this lesson. In this lesson, we learned that
inscribed angles subtended by the same arc are equal. We also saw that inscribed angles
subtended by congruent arcs or arcs of equal measure are also equal. Finally, we learned that the
reverse of these ideas is also true. If there are two congruent angles
subtended by the same line segment and on the same side of that line segment, then
their vertices and the endpoints of that line segment lie on a circle. And in that circle, the line
segment is a chord.
