The correct answer is A) 112.5°. The compass course of a vessel heading ESE (East-Southeast) is 112.5°. This is because on a standard compass rose, the east (E) direction is 90° and the southeast (SE) direction is 135°. The midpoint between these two cardinal directions is 112.5°, which represents the compass course for a vessel heading ESE. The other options are incorrect because B) 135.0° represents the compass course for a vessel heading directly southeast (SE), C) 157.5° is too far to the south, and D) 180.0° represents a vessel heading due south, which is not the same as ESE.