angle A equals 30 degrees. arc AC is equal to 2x angle B, making arc AC equal 120 degrees. arc AB equals 180 degrees. since it is half of the circle. this means that arc BC must equal 60 degrees, because the final arc must add up to 360 with the other arcs. 360 - ( 180 + 120 ) = 60 degrees. since angle A faces arc BC, angle A must equal half of arc BC. 60 divided by 2 = 30, making angle A equal 30 degrees.