O is the circumcentre of triangle ABC and ∠BAC = 55°. What is the measure of ∠OBC?

The angle at the centre is twice the angle at the circumference standing on the same arc BC, so ∠BOC = $2\times 55$° = 110°. OB and OC are radii, so triangle OBC is isosceles and ∠OBC = (180° − 110°)/2 = 35°. In general ∠OBC = 90° − A. Answering 110° stops at the central angle, and 27.5° halves the vertex angle as if O were the incentre.