Consider the given circle. Find the measure of the arc 𝐶𝐷.
We recall first that the arc 𝐶𝐷 means the minor arc between points 𝐶 and 𝐷. That’s this arc here on the diagram. We recall also that the measure of an arc is the measure of its central angle. That’s the angle between the two radii connecting the endpoints of the arc to the center of the circle. For the arc 𝐶𝐷, the two radii are the line segments 𝐶𝑀 and 𝐷𝑀. So the central angle for this arc is angle 𝐶𝑀𝐷. If we want to determine the measure of the arc 𝐶𝐷 then, we need to determine the measure of the angle 𝐶𝑀𝐷.
Now, in the figure, we’re given the measures of two other angles: angle 𝐵𝑀𝐶, which is 56 degrees, and angle 𝐴𝑀𝐷, which is 32 degrees. Now, as the line segment 𝐴𝐵 is a diameter of the circle, so it is a straight line, the measures of the three angles 𝐵𝑀𝐶, 𝐶𝑀𝐷, and 𝐴𝑀𝐷 must sum to 180 degrees. We can therefore form an equation. 56 degrees plus the measure of angle 𝐶𝑀𝐷 plus 32 degrees equals 180 degrees. We can find the measure of angle 𝐶𝑀𝐷 by subtracting 56 degrees and 32 degrees from each side of the equation. The measure of angle 𝐶𝑀𝐷 is 180 degrees minus 56 degrees minus 32 degrees, which is 92 degrees. Angle 𝐶𝑀𝐷 is the central angle for the arc 𝐶𝐷. So the measure of the arc 𝐶𝐷 is also 92 degrees.