In the above figure, point O is at the center of the circle, and points A, B and C are on the circumference of the circle. If ∠BCE = 3xº, what is the measure of ∠ABO? A) (x + 10)º B) (180 - x)º C) (90 - x)º D) (2x - 30)º E) (3x - 90)º Source: GMAT Prep Now Difficulty level: 700 Answer: E

Since BCE = 3X, its supplementary angle BCA = 180-3X. Since BCA and AOB subtend the same chord, and one is an inscribed angle and the other, central, AOB = 2 x BCA, So AOB = 2(180-3X) = 360-6X Since AO and BO are equal, being radii, OAB and ABO are equal Since sum of angles = 180, therefore 2 x ABO + 360-6X = 180, and 6x - 180 = 2 x ABO, therefore ABO = 3X-90

Brent@GMATPrepNow wrote: In the above figure, point O is at the center of the circle, and points A, B and C are on the circumference of the circle. If ∠BCE = 3xº, what is the measure of ∠ABO? A) (x + 10)º B) (180 - x)º C) (90 - x)º D) (2x - 30)º E) (3x - 90)º Given: Since angles on a line must add to 180 degrees, we know that ∠ACB = (180 - 3x) degrees Since the inscribed angle ∠ACB and the central angle ∠AOB both contain (hold) the same chord (AB), we know that the central angle is TWICE the inscribed angle. In other words, ∠AOB = 2(180 - 3x) = 360 - 6x Finally, since OA and OB are radii, we know that these lengths are EQUAL If these lengths are EQUAL, then ∆AOB is an ISOSCELES triangle, and ∠OAB = ∠ABO Let q = ∠OAB = ∠ABO Since all 3 angles in ∆AOB add to 180 degrees, we can write: (360 - 6x) + q + q = 180 Rearrange: 2q = 6x - 180 Divide both sides by 2 to get: q = 3x - 90 In other words, ∠ABO = 3x - 90 Answer: E

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.