The length of a circular arc (s) is the product of the circle's radius and the central angle measure in radians.
... s = r·θ
Then the central angle measure in radians is ...
... θ = s/r
For your problem, you have s = 8π/9 and r = 1. Then the central angle AOC is
... ∠AOC = (8π/9)/(1) = 8π/9 . . . . radians
The relationship between radians and degrees is
... 180° = π radians
Multiplying this equation by 8/9 will tell us the degree measure of ∠AOC.
... 8/9×180° = (8/9)×π radians
... ∠AOC = (8/9)·180° = 160°
We know that this is the sum of the two (equal) central angles ∠AOB and ∠BOC, so we have
... 2×∠AOB = 160°
... ∠AOB = 160°/2 = 80°
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Here, you know that 8π/9 is the central angle in radians because the ratio of arc length to radius is the angle in radians.