Final answer:
The correct word to describe the complex reasoning is 'convoluted.' In Euclidean geometry, a straight line is the shortest distance between two points, but in non-Euclidean geometries, the shortest path is a geodesic.
Step-by-step explanation:
The correct word to complete the student's sentence about the reasoning used to "prove" that a straight line is not necessarily the shortest distance between two points is convoluted. The word convoluted implies that the reasoning was complicated and difficult to follow, which aligns with the student's difficulty in understanding the proof involved.
It's important to note that, in Euclidean geometry, a straight line is indeed the shortest distance between two points. However, in non-Euclidean geometries, such as those applied in general relativity where space is curved, the shortest path between two points is known as a geodesic, which may not appear "straight" in the conventional sense.