AOC= EXTERIOR ANGLE = 90+30=120 or Linear pair

1. Tangent information tells us that the line is perpendicular, thus has a 90 degree angle for AOB.

2. 30 +90 means the third angle (either by 180-120 or from recognizing a 30-60-90) is 60 (which is angle AOB).

3. All lines on the GRE are assumed straight (180 degrees). Since we have already determined that angle AOB is 60, we know angle AOC is equal to 180-60...so 120.

4. The arc angle being asked for matches the angle opposite to it, so if AOC=120 then Arc AC=120 as well.

5. Quantity A asked for half of Arc AC, so 120/2 = 60.
Quantity B asked for angle AOB, so 60.

6. 60=60, so the answer is C, Quantity A and quantity B are equal.

Does the widehat above AC in quantity A represent the central angle or the arc length of AC?

Bit confused. If it is the arc length, then the answer is D, but if is the central angle, then it is C.

I bet a lot of people are getting this question wrong merely b/c of notation. As it is quantity A is using the notation of the arc length of AC and not the measure of the central angle of arc AC.

So if seen as arc length then it would depend on the radius and the answer would be D

Another alternative:

You can use the secant relationship:

\(\frac{\widehat{AC}- \angle AOB}{2} = \angle OBA\)

We know that \(\angle OBA = \frac{\angle OAB}{2 }\) since it's a 30-60-90 triangle.

\(\frac{\widehat{AC}- \angle AOB}{2} = \frac{\angle AOB}{2}\)

\(\frac{\widehat{AC}}{2} = \angle AOB \)