Lesson Video: Inscribed Angles Subtended by the Same Arc Mathematics • Third Year of Preparatory School

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.