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.