Question

Which of the following best compares parallel lines in Euclidean and spherical geometry?

A. In Euclidean geometry, for any line and point not on the line, there exist two parallel lines that pass through the point. In spherical geometry, for any line and point not on the line, no parallel line exists.

B. In Euclidean geometry, for any line and point not on the line, there exists one parallel line that passes through the point. In spherical geometry, for any line and point not on the line, there exist two parallel lines that pass through the point.

C. In Euclidean geometry, for any line and point not on the line, there exists one parallel line that passes through the point. In spherical geometry, for any line and point not on the line, no parallel line exists.

D. In Euclidean geometry, for any line and point not on the line, no parallel line exists. In spherical geometry, for any line and point not on the line, there exists one parallel line that passes through the point.

Answer 1

In Euclidean geometry, there is one parallel line through a point not on a line, while in spherical geometry no parallel lines exist for a point not on a line.

Step-by-step explanation:

The question asks to compare parallel lines in Euclidean geometry and spherical geometry. In Euclidean geometry, for any given line and a point not on the line, there is exactly one line that passes through the point which is parallel to the original line. This is due to Euclid's parallel postulate which states that through any point not on a line, there is exactly one line that does not intersect the original line. On the other hand, in spherical geometry, for any line and a point not on the line, no parallel lines exist because all lines (great circles) on a sphere eventually intersect. Therefore, the correct answer to the comparison is option C: In Euclidean geometry, for any line and point not on the line, there exists one parallel line that passes through the point. In spherical geometry, for any line and point not on the line, no parallel line exists.