Given that line segment 𝐴𝐵 is a
diameter in circle 𝑀 and the measure of angle 𝐵𝑀𝐷 equals 59 degrees, find the
measure of angle 𝐴𝐶𝐷 in degrees.
Let’s put what we know into the
diagram. Angle 𝐵𝑀𝐷 measures 59 degrees,
and we’re trying to find the measure of angle 𝐴𝐶𝐷. If we start with what we know about
angle 𝐵𝑀𝐷, since 𝐵𝑀𝐷 has a vertex at the center of the circle, 𝐵𝑀𝐷 is a
central angle. And because angle 𝐵𝑀𝐷 is a
central angle, its subtended arc, arc 𝐵𝐷, also measures 59 degrees. We’re also interested in angle
𝐴𝐶𝐷. But angle 𝐴𝐶𝐷 is not a central
angle. It’s an inscribed angle because its
vertex is on the circumference of the circle, as are both endpoints.
The arc associated with angle
𝐴𝐶𝐷 would be arc 𝐴𝐷. We have a partial measurement for
this arc, but we’re missing the distance from 𝐴 to 𝐵. But because we know that 𝐴𝐵 is a
diameter, it cuts the circle in half. And that means the measure of arc
𝐴𝐵 is 180 degrees. If arc 𝐴𝐵 equals 180 and arc 𝐵𝐷
equals 59 degrees, we can say the measure of arc 𝐴𝐷 is equal to the measure of arc
𝐴𝐵 plus the measure of arc 𝐵𝐷.
If we plug in what we know, the
measure of arc 𝐴𝐷 is 239 degrees. Because angle 𝐴𝐶𝐷 is an
inscribed angle and it has a subtended arc measure of 239 degrees, we can find out
the exact measure of angle 𝐴𝐶𝐷. The measure of the inscribed angle
𝐴𝐶𝐷 will be one-half its subtended arc, arc 𝐴𝐷. Since that arc is 239 degrees, we
take half of that and we get 119.5 degrees for the measure of angle 𝐴𝐶𝐷.