In a circle with centre O, let AB be a diameter and P be a point on the circumference. If the central angle POA is 120 degrees with vertex at O, what is the measure in degrees of angle PBO?

Introduction / Context:
This geometry question involves angles in a circle with a diameter and a central angle. It tests understanding of the relationships between central angles, points on the circle and angles formed at other points on the circle. Such questions train you to visualise circle geometry and use symmetry effectively.

Given Data / Assumptions:
- AB is a diameter of the circle with centre O.
- P is a point on the circumference of the circle, different from A and B.
- The central angle POA measures 120 degrees at the centre O.
- We need to find the measure of angle PBO, which is the angle at point B between segments BP and BO.

Concept / Approach:
One clean approach is to use coordinate geometry. Place the circle with centre at the origin, take the radius as 1, and put A and B conveniently on the x axis. Then interpret the angle POA = 120 degrees as giving the direction of OP. After that, compute the angle at B using vectors or geometric reasoning. The result is independent of the specific radius choice.

Step-by-Step Solution:
Place the circle with centre O at (0, 0) and radius 1. Let A be at (1, 0) and B at (-1, 0), so AB is a diameter. The angle POA = 120 degrees means that OP makes an angle of 120 degrees with OA, which lies along the positive x axis. Thus, the coordinates of P are (cos 120°, sin 120°) = (-1/2, √3/2). Now consider angle PBO at B formed by vectors BP and BO. Vector BP = P - B = (-1/2 + 1, √3/2 - 0) = (1/2, √3/2). Vector BO = O - B = (0 + 1, 0 - 0) = (1, 0). The angle between these two vectors is 60 degrees because BP lies at 60 degrees to the positive x axis while BO points along the positive x axis.

Verification / Alternative check:
A purely geometric argument also works. Since AB is a diameter, triangle AOB is a straight line at the centre. Rotating OP 120 degrees from OA places P symmetrically in such a way that triangle BOP forms a 60 degree angle at B due to the known angles in equilateral type configurations. Either method yields 60 degrees.

Why Other Options Are Wrong:
Angles 30, 40, 45 and 50 degrees do not match the strict symmetry of this configuration. They usually come from misinterpreting which angle is 120 degrees or confusing central angles with inscribed angles that subtend the same arc.

Common Pitfalls:
A frequent mistake is to assume that angle PBO equals half of 120 degrees directly, which is not true here because PBO is not an inscribed angle subtending the same arc as POA. Another pitfall is misplacing the points around the circle mentally and mixing up which angle is at the centre versus at the circumference.

Final Answer:
The measure of angle PBO is 60°.