In a circle, one angle subtended by a diameter at any point on the semicircle is asked. What is the measure (in degrees) of any angle in a semicircle?

Introduction / Context:
This problem tests a fundamental property of circles related to angles formed by diameters. When an angle is subtended by a diameter of a circle at a point on the circle, that angle always lies in a semicircle. The question asks for the measure of such an angle, often referred to as the angle in a semicircle. This is a standard result in circle geometry and is frequently used in many competitive examinations and school level problems.

Given Data / Assumptions:
- We have a circle with a diameter drawn.- A point is chosen on the circumference forming a triangle with the endpoints of the diameter.- The angle at this point on the semicircle is called the angle in a semicircle.- The circle is a standard Euclidean circle with usual geometric properties.

Concept / Approach:
The important theorem here is: “The angle in a semicircle is a right angle.” In other words, if a triangle is drawn in a circle such that one side of the triangle is the diameter of the circle, then the angle opposite this diameter is always 90 degrees. This result comes from the properties of cyclic quadrilaterals or can be derived using the concept of central angles and inscribed angles. The measure of an inscribed angle is half the measure of the central angle subtending the same arc.

Step-by-Step Solution:
Step 1: Consider a circle with centre O and diameter AB.Step 2: Take a point C on the circle such that A, B, and C form a triangle ABC.Step 3: Since AB is a diameter, the central angle AOB subtending arc ACB is 180 degrees (a straight line).Step 4: The angle at the circumference subtending the same arc ACB is angle ACB, which is the angle in the semicircle.Step 5: By the inscribed angle theorem, the measure of an inscribed angle is half of the corresponding central angle.Step 6: Therefore, angle ACB = 180 degrees / 2 = 90 degrees.Step 7: Hence, any angle in a semicircle is a right angle.

Verification / Alternative check:
An alternative view is to think of triangle ABC as a right angled triangle with AB as the hypotenuse. This corresponds with the converse of the theorem stating that the locus of all points forming a right angle with a given segment as hypotenuse is a circle with that segment as diameter. Both results point to the same conclusion: the angle in a semicircle is always 90 degrees, and this is independent of the position of the third point C on the semicircle.

Why Other Options Are Wrong:
- 45°: This angle is too small and does not satisfy the semicircle property for all such points.- 60°: This angle would correspond to an equilateral triangle under special conditions, which is not guaranteed here.- 120°: This is an obtuse angle and cannot be the angle in a semicircle according to the established theorem.- 150°: This is far too large and does not correspond to the fixed relationship between the diameter and inscribed angle.

Common Pitfalls:
Some learners confuse angles in semicircles with other special angles like 60 degrees or 45 degrees which appear in equilateral or isosceles right triangles. Another common mistake is to think the angle value may vary depending on the position of the third point on the semicircle, but the theorem clearly states that it is always a right angle. Remembering that a diameter subtending an angle at the circle always produces a right angle helps avoid such confusion.

Final Answer:
The measure of any angle in a semicircle is 90°.