In circle geometry, O is the centre of the circumcircle of triangle ABC, and the point A and the chord BC lie on opposite sides of O. If the central angle ∠BOC is 150 degrees, find the measure of the inscribed angle ∠BAC in degrees.

Introduction / Context:
This question checks understanding of the relationship between central angles and inscribed angles in a circle. When a triangle is inscribed in a circle, each angle at the circumference corresponds to a central angle subtended by the same chord. Mastery of this relationship is very important in coordinate geometry, Euclidean geometry, and many entrance exam problems involving circles and cyclic quadrilaterals.

Given Data / Assumptions:
- O is the centre of the circumcircle of triangle ABC.
- Chord BC and vertex A are on the circle, and A and BC lie on opposite sides of O, ensuring a proper triangle configuration.
- The central angle ∠BOC is given as 150 degrees.
- We must find the inscribed angle ∠BAC, which also subtends chord BC.

Concept / Approach:
The key theorem is that an inscribed angle in a circle is half the measure of the central angle subtending the same chord. Here, both the central angle ∠BOC and the inscribed angle ∠BAC stand on chord BC. Therefore, ∠BAC should be exactly half of ∠BOC, as long as they are on the same arc of the circle. This is a direct application question with no additional construction needed.

Step-by-Step Solution:
1) Identify that chord BC subtends the central angle ∠BOC at the centre O. 2) Recognise that the same chord BC subtends the inscribed angle ∠BAC at the point A on the circumference. 3) Apply the theorem: inscribed angle = half of the central angle standing on the same chord. 4) Compute ∠BAC = (1/2) * ∠BOC = (1/2) * 150 degrees. 5) Therefore, ∠BAC = 75 degrees.

Verification / Alternative check:
We can reason in reverse: if an inscribed angle were 75 degrees, then the central angle standing on the same chord must be 150 degrees. This is consistent with the given data. Additionally, this result does not depend on the precise shape of triangle ABC, only on the circle property, so any configuration with chord BC and vertex A on the circle satisfying the condition will always give the same angle value.

Why Other Options Are Wrong:
Option B (70 degrees), Option D (65 degrees), and Option E (80 degrees) are simply numbers near the correct result, often used to trap students who attempt rough estimates without applying the exact half relation. Option C (60 degrees) might come from confusing the given angle with an interior triangle angle of a regular hexagon or other special figure, but it does not match half of 150 degrees. Only Option A matches one half of the central angle exactly.

Common Pitfalls:
A frequent error is to confuse which angle is central and which is inscribed, leading to doubling instead of halving, or to using the wrong chord. Some learners also mistakenly think that the inscribed angle is always half of any angle in the diagram rather than specifically the central angle on the same chord. Drawing a quick sketch and marking the chord BC and vertex A clearly can prevent such misunderstandings.

Final Answer:
Using the circle theorem that an inscribed angle is half the corresponding central angle, we get ∠BAC = 150° / 2 = 75°.