The question in the image is asking which statement best compares parallel lines in Euclidean and spherical geometry. Let's analyze the concepts:

1. Euclidean Geometry (normal flat-plane geometry) follows the Parallel Postulate, which says that for any given line and a point not on that line, there is exactly one line parallel to the original line passing through the point.

2. Spherical Geometry involves the surface of a sphere. In this geometry, the concept of parallel lines does not exist in the same way as in Euclidean geometry. For any two lines (great circles), they eventually intersect.

Let's break down the choices:

• Option A: Claims that in Euclidean geometry, two parallel lines can pass through a point not on the line. This is incorrect because there is only one parallel line through a point in Euclidean geometry. The statement about spherical geometry is correct, as no parallel lines exist in spherical geometry.

• Option B: Incorrect because in Euclidean geometry, there is only one parallel line, not two.

• Option C: Correct. It states that in Euclidean geometry, no parallel line exists through the point in spherical geometry, which is accurate because lines (great circles) intersect in spherical geometry.

• Option D: Incorrect because the first part suggests that no parallel line exists in Euclidean geometry, which contradicts the Euclidean Parallel Postulate.

Thus, Option C is the correct answer.

Would you like a more detailed explanation? Here are five questions to deepen your understanding:

1. What is the difference between Euclidean and non-Euclidean geometries?
2. How does the Parallel Postulate apply in different geometries?
3. What are "great circles" in spherical geometry?
4. Why can’t parallel lines exist in spherical geometry?
5. Can you provide an example of non-Euclidean geometries besides spherical geometry?

Tip: In Euclidean geometry, the angles of a triangle always add up to 180°, but this is not true in non-Euclidean geometries like spherical geometry.