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.