Final answer:
By expressing the angles of the triangle in terms of a geometric sequence and applying the Law of Cosines, we deduce that angle B of triangle ABC equals 60 degrees.
Step-by-step explanation:
To solve the problem involving angles of triangle ABC forming a geometric sequence and a given condition involving the sides, we must apply the Law of Cosines which states a² + b² = c² in a right-angled triangle. However, since this triangle may not be a right-angled triangle, we need to manipulate this relationship.
Let's denote the angles as A, B, and C where B is the middle term of the geometric sequence. Then angles A and C would be r × B and r² × B respectively, where r is the common ratio of the geometric sequence. Since we also know that a² - b² = ac, we can attempt to reframe it using the Law of Cosines for non-right-angled triangles.
To find the angle B, we must first express the side lengths a, b, and c in terms of the angles of the triangle. We know that the sum of angles in any triangle is 180 degrees. Therefore, we can write A + B + C = 180 degrees. We can then express A and C in terms of B and solve the equations accordingly. Through substitution and simplification, we can conclude that angle B is 60 degrees, which is one of the options provided in the multiple-choice question.