A straight line touches a circle at the point A, and AB is a chord of the circle. The angle between the tangent and the chord AB measures 58°. What angle does the chord AB subtend at the centre of the circle, measured on the side of the alternate segment?

By the alternate-segment theorem, the tangent–chord angle equals the angle that AB subtends at any point of the alternate segment, so that inscribed angle is 58°. The angle at the centre is twice the angle at the circumference standing on the same arc, so the central angle is $2\times 58$° = 116°. Halving instead of doubling gives 29°, and taking the supplement gives 122°.