Final answer:
The information provided on circle theorems and arc lengths is insufficient to calculate the size of angle x without a specific diagram or additional context. Common circle theorems cannot be applied to determine the correct angle size based on the given text.
Step-by-step explanation:
The question seems to involve circle theorems and arc lengths, but the information provided is not sufficient to work out the size of angle x using the available theorems. The details given in the text, such as arc length, radius of curvature, and angle of rotation, are related to understanding how to calculate the length of an arc or the size of an angle in a circle, based on the relationship between angles and arcs in a circular setting. However, without a specific diagram or further information, it's impossible to provide a calculation or theorem application for angle x.
Generally, in problems like this, you might apply theorems like the angle at the center theorem, which states that the angle subtended at the center of the circle is twice the angle subtended at the circumference by the same arc, or the inscribed angle theorem, which states that an angle inscribed in a circle is half the measure of its intercepted arc.
Without the actual diagram or additional context, we cannot determine the correct size of angle x or utilize other common circle theorems such as tangent-secant theorem, alternate segment theorem, or cyclic quadrilateral theorem.