Video: Finding the Measure of an Inscribed Angle given Its Arc’s Measure by Solving Two Linear Equations

Video Transcript

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.