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O is the center of the circle above, and lines AO and DC are parallel. If angle AOB has a measure of 50°, what is the measure of minor arc CD?

Correct Answer:

**C**An angle inscribed into a circle is half of the arc it traces out.

- To understand this problem, it is helpful to know that any inscribed angle in a circle creates a minor arc that has twice the measure of the angle. So here, we have angle DCB which is exactly 1/2 the measure of minor arc DB.
- To determine the measure of angle DCB, notice that lines AO and DC are parallel to one another, intersecting with the same line, CB. Thus, the interior angles formed from the intersection of line DC and CB, and the intersection of lines AO and CB must be the same. Therefore, angle DCB must also have a measure of 30°.
- Now we can determine the measure of minor arc DB, because we know, as stated in step 1, that a minor arc has a measure that is twice that of its inscribed angle. Therefore, minor arc DB must have a measure of 100°, which is twice that of 50°.
- Note that line CB runs through O, the center of the circle. This means that line CB must be the diameter of this circle, because any line that runs through the center of the circle and connects with the perimeter of the circle is a diameter.
- Diameters necessarily cut circles into half. Given that the measurement of a circle is 360°, we know that the measurement of half the circle is 180° Therefore arc CB has a measurement of 180°
- We know that arc DB has a measurement of 100°, so 180 minus 100 must equal the measurement of minor arc CD. In equation form, 180 – 100 = arc CD. So, the measurement of minor arc CD must be 80°, answer choice (C).