AB is a chord of a circle with center O, C is the midpoint of the chord AB, extend OC to the point D on the circle. If OC=CD, what is the measure of angle AOB?

90
100
120
60

We can't see the point D.

To find the measure of angle AOB, we need to use a property of circles called the Inscribed Angle Theorem.

According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of its intercepted arc.

In this case, angle AOB is the inscribed angle, and the intercepted arc is arc AB. Since C is the midpoint of chord AB, we know that the intercepted arc AB is divided equally into two arcs: AC and CB.

Given that OC = CD, we can conclude that angles OCA and CDO are congruent because they are opposite angles formed by intersecting chords. This means that arcs AC and CB are also congruent.

Since arcs AC and CB are congruent and they add up to the intercepted arc AB, each arc must measure half of the intercepted arc, which is AB.

Therefore, the measure of angle AOB is equal to the measure of arc AB.

Now let's examine the answer choices:
90 degrees: This is not possible because angle AOB is equal to the measure of arc AB, and a circle has 360 degrees. So if angle AOB were 90 degrees, the measure of arc AB would be 90 degrees as well, which is not possible since it would not cover the whole circle.

100 degrees: This is also not possible because, as mentioned before, the measure of angle AOB is equal to the measure of arc AB, and a circle has 360 degrees. So if angle AOB were 100 degrees, the measure of arc AB would be 100 degrees as well, which is not possible since it would not cover the whole circle.

120 degrees: This is the correct answer. Since angle AOB is equal to the measure of arc AB, which covers one-third of the circle, angle AOB must also measure one-third of 360 degrees, which is 120 degrees.

60 degrees: This is incorrect. If angle AOB were 60 degrees, the measure of arc AB would be 60 degrees as well, which is too small and would not cover the whole circle.

Therefore, the measure of angle AOB is 120 degrees.