Question Video: Finding the Measure of an Inscribed Angle given the Measure of Another Circle’s Inscribed Angle Involving Concentric Circles Mathematics

Video Transcript

In the figure, line segments 𝐴𝐸 and 𝐵𝐶 pass through the center of the circles. Given that the measure of angle 𝐹𝐸𝐷 equals 50 degrees and the measure of angle 𝐶𝐵𝐴 is equal to two 𝑥 minus 10 degrees, find 𝑥.

We are told in the question that line segments 𝐴𝐸 and 𝐵𝐶 pass through the center of the circles and that the measure of angles 𝐹𝐸𝐷 and 𝐶𝐵𝐴 are 50 degrees and two 𝑥 minus 10 degrees, respectively.

We begin by recalling that angles subtended from the same arc are equal. And we also know that angles subtended from arcs with equal measure are equal. This is really useful when we’re working with a pair of concentric circles as in this question, as we’re able to say that the measure of arc 𝐹𝐷 is equal to the measure of arc 𝐶𝐴. And they are both equal to the measure of the central angle shown.

Since the measure of those two arcs are equal, then the measure of any angle 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. This means that 50 degrees is equal to two 𝑥 minus 10 degrees. And 50 must be equal to two 𝑥 minus 10.

We can now solve this equation for 𝑥 by firstly adding 10 to both sides. This gives us 60 is equal to two 𝑥. We can then divide through by two such that 30 is equal to 𝑥 or 𝑥 is equal to 30.