Lesson Video: Inscribed Angles in a Circle Mathematics • Third Year of Preparatory School

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, let’s remember what an inscribed angle is. It’s 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 𝑎 degrees, then the arc created between these two endpoints will be two 𝑎 degrees. Another way to say that is an inscribed angle measures half of the subtended arc that’s created by that angle. If we have another inscribed angle, which has the same endpoints as the first one, this angle will also measure 𝑎 degrees because 𝑎 is one-half of the arc that’s created by these endpoints, which here is two 𝑎.

It’s 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 circle’s 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 don’t 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 we’re dealing with a central angle, its angle measure will be equal to the subtended arc’s angle measure. The way we’ve drawn it here, it will be equal to two 𝑎. 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 𝐴𝐷 will be equal to arc 𝐵𝐶. That is to say, in a circle, arcs between parallel chords are congruent. In the circle we’ve drawn here, the measure of arc 𝐴𝐷 will be equal to the measure of arc 𝐵𝐶. We’re now ready to use these circle theorems to calculate unknown angles.

In the figure, 𝑂 is the center and the measure of angle 𝑂𝐴𝐵 equals 59.5 degrees. What is the measure of angle 𝐴𝑂𝐵? What is the measure of angle 𝐴𝐶𝐵?

We’re told that 𝑂 is the center of this circle and that the measure of angle 𝑂𝐴𝐵 is 59.5 degrees. We want to find the measure of angle 𝐴𝑂𝐵 and the measure of 𝐴𝐶𝐵. We see that the points 𝐴, 𝑂, and 𝐵 form a triangle. Both line segment 𝑂𝐵 and line segment 𝑂𝐴 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 𝑂𝐴 is equal to line segment 𝑂𝐵. And it will mean that triangle 𝐴𝑂𝐵 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 𝐴𝐵𝑂 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 𝑂𝐴𝐵 and angle 𝐴𝐵𝑂. We add the two angles we know, and we get 119 degrees. And then to solve for angle 𝐴𝑂𝐵, we subtract 119 degrees from both sides, and we find that angle 𝐴𝑂𝐵 is equal to 61 degrees. That’s the answer to part one.

Part two is a little bit less straightforward. We notice that both of these angles share the endpoints 𝐴, 𝐵, which means they’re both subtended by the arc 𝐴𝐵. But we need to make a clarification here. Angle 𝐴𝑂𝐵 is a central angle that is subtended by arc 𝐴𝐵, while angle 𝐴𝐶𝐵 is an inscribed angle subtended by arc 𝐴𝐵. 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 𝑎, the inscribed angle subtended by the same points will be 𝑎 degrees.

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

Here’s another example.

From the figure, what is 𝑥?

Let’s start with what we know. We have angle 𝐴𝐶𝐵, which measures 101 degrees. And we have angle 𝐴𝑀𝐵. In this case, we’re talking about the reflex of angle 𝐴𝑀𝐵. That’s the one that’s greater than 180 degrees, which measures two 𝑥 plus eight degrees. Angle 𝐴𝐶𝐵 and angle 𝐴𝑀𝐵 share the endpoints 𝐴 and 𝐵. But because the vertex of angle 𝐴𝑀𝐵 is the center of the circle, we say that angle 𝐴𝑀𝐵 is a central angle for this circle. While the vertex for angle 𝐴𝐶𝐵 is on the outside of the circle, making angle 𝐴𝐶𝐵 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 𝑎 degrees, and that would make the central angle two 𝑎 degrees. By this, we can say that the measure of angle 𝐴𝑀𝐵 is going to be equal to two times the measure of angle 𝐴𝐶𝐵. 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 𝑥 plus eight will be equal to two times 101. When we multiply two times 101, we get 202. And now, we’re ready to solve for 𝑥. Subtract eight from both sides. Two 𝑥 equals 194. Then, divide both sides by two, and we find that 𝑥 equals 97.

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

Given that the measure of angle 𝐴𝐵𝐷 equals 44 degrees and the measure of angle 𝐶𝐸𝐴 equals 72 degrees, find 𝑥, 𝑦, and 𝑧.

Let’s start by listing what we know. Angle 𝐴𝐵𝐷 equals 44 degrees, angle 𝐶𝐸𝐴 measures 72 degrees. These two chords intersect at point 𝐸. And that means we can say that angle 𝐵𝐸𝐷 and angle 𝐶𝐸𝐴 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 𝐵𝐸𝐷 is also equal to 72 degrees. The points 𝐸, 𝐵, and 𝐷 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 𝑧 equals 180. So, we subtract 116 from both sides. And we find that 𝑧 equals 64 degrees.

We won’t be able to follow the same procedure to find 𝑥 and 𝑦. So, we’ll need to think about some of the circle theorems. If we look at inscribed angle 𝐵, we see that it has endpoints along the circle at 𝐴 and 𝐷 and that its intercepted arc is arc 𝐴𝐷. We could write them as arc 𝐴𝐷 intercepts angle 𝐴𝐵𝐷. But there’s another angle in this circle that also intercepts the same arc, and that would be angle 𝐴𝐶𝐷. Because both of these angles are subtended by the same arc, we can say that the measure of angle 𝐴𝐶𝐷 will be equal to the measure of angle 𝐴𝐵𝐷. And that means 𝑥 will be equal to 44 degrees.

And because all three angles need to sum to 180 degrees, we can tell that angle 𝑦 is going to be equal to 64 degrees. If we wanted to confirm this, we could see that angle 𝐶𝐴𝐵 intercepts arc 𝐶𝐵 and angle 𝐶𝐷𝐵 intercepts arc 𝐶𝐵. And so, we found that 𝑥 equals 44 degrees, and both 𝑦 and 𝑧 equals 64 degrees.

In our next example, we’ll have a diameter to consider.

Given that line segment 𝐴𝐵 is a diameter in circle 𝑀 and the measure of angle 𝐵𝑀𝐷 equals 59 degrees, find the measure of angle 𝐴𝐶𝐷 in degrees.

Let’s put what we know into the diagram. Angle 𝐵𝑀𝐷 measures 59 degrees, and we’re trying to find the measure of angle 𝐴𝐶𝐷. If we start with what we know about angle 𝐵𝑀𝐷, since 𝐵𝑀𝐷 has a vertex at the center of the circle, 𝐵𝑀𝐷 is a central angle. And because angle 𝐵𝑀𝐷 is a central angle, its subtended arc, arc 𝐵𝐷, also measures 59 degrees. We’re also interested in angle 𝐴𝐶𝐷. But angle 𝐴𝐶𝐷 is not a central angle. It’s an inscribed angle because its vertex is on the circumference of the circle, as are both endpoints.

The arc associated with angle 𝐴𝐶𝐷 would be arc 𝐴𝐷. We have a partial measurement for this arc, but we’re missing the distance from 𝐴 to 𝐵. But because we know that 𝐴𝐵 is a diameter, it cuts the circle in half. And that means the measure of arc 𝐴𝐵 is 180 degrees. If arc 𝐴𝐵 equals 180 and arc 𝐵𝐷 equals 59 degrees, we can say the measure of arc 𝐴𝐷 is equal to the measure of arc 𝐴𝐵 plus the measure of arc 𝐵𝐷.

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

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

Given that the line segment 𝐴𝐵 is a diameter of the circle and line segment 𝐷𝐶 is parallel to line segment 𝐴𝐵, find the measure of angle 𝐴𝐸𝐷.

We’re interested in the measure of angle 𝐴𝐸𝐷; that’s this measure. And we’ve been given a few other pieces of information. We know line segment 𝐷𝐶 is parallel to line segment 𝐴𝐵. We know line segment 𝐴𝐵 is the diameter. And on the figure, angle 𝐶𝐵𝐴 has been labeled as 68.5 degrees.

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

We know 𝐴𝐵 needs to be 180 degrees and arc 𝐶𝐴 is 137 degrees. To solve for the measure of arc 𝐵𝐶, we can subtract 137 from both sides of the equation. And we get the measure of arc 𝐵𝐶 is 43 degrees. And here’s 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 𝐶𝐵 equals 43 degrees, arc 𝐷𝐴 also equals 43 degrees.

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

Before we finish, let’s quickly review the key points. If you have a central angle that measures two 𝑎 degrees, its intercepted arc will also measure two 𝑎 degrees. While an inscribed angle that intercepts the same arc will have half the angle measure, only 𝑎 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.