Let x = 55, implying that 3x=155.
Since ∠BCE and ∠ACB must sum to 180, we get:
An INSCRIBED angle is formed by two chords.
A CENTRAL angle is formed by two radii.
When an inscribed angle and a central angle intercept the same two points on a circle, the central angle is twice the inscribed angle.
Since inscribed angle ∠ACB and central ∠AOB both intercept points A and B on the circle, ∠AOB must be twice ∠ACB, implying that ∠AOB = 30:
The angles inside of ∆AOB must sum to 180.
Since OA and OB are radii -- and thus are equal -- the angles opposite OA and OB (∠OAB and ∠ABO) must also be equal.
The result is the following figure:
The question stem asks for the value of ∠ABO: 75.
This is our target.
Now plug x=55 into the answers to see which yields our target of 75.
Only
E works:
3x - 90 = (3*55) - 90 = 75.
The correct answer is
E.
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