Let's break down the image question:

You are asked to identify which of the following statements are true for Euclidean geometry or spherical geometry. Here's a detailed explanation of each:

1. Parallel lines do not intersect each other:

   • Euclidean Geometry: True. In Euclidean geometry, parallel lines, by definition, never intersect.
   • Spherical Geometry: False. In spherical geometry, "straight lines" (geodesics) are great circles. Any two great circles on a sphere intersect at two points.

   So, the correct answer here is Euclidean Geometry.

2. Two straight lines may intersect at no more than one point:

   • Euclidean Geometry: True. In Euclidean geometry, two distinct straight lines can intersect at exactly one point.
   • Spherical Geometry: False. As mentioned before, in spherical geometry, two great circles intersect at two points (the ends of a diameter).

   So, the correct answer is Euclidean Geometry.

3. The sum of the measures of the interior angles of a triangle can vary:

   • Euclidean Geometry: False. In Euclidean geometry, the sum of the interior angles of a triangle is always 180°.
   • Spherical Geometry: True. In spherical geometry, the sum of the interior angles of a triangle exceeds 180°, and the exact value depends on the area of the triangle.

   So, the correct answer here is Spherical Geometry.

Summary of answers:

1. Euclidean Geometry for "Parallel lines do not intersect each other".
2. Euclidean Geometry for "Two straight lines may intersect at no more than one point".
3. Spherical Geometry for "The sum of the measures of the interior angles of a triangle can vary".

Do you want more details or explanations on any of these geometries or concepts?

Five related questions:

1. What are the key differences between Euclidean and spherical geometry?
2. How does the concept of parallel lines differ in non-Euclidean geometries?
3. Can you give an example of a real-world application of spherical geometry?
4. How are angles measured differently in spherical triangles compared to Euclidean ones?
5. What are geodesics, and how do they behave in spherical versus Euclidean geometry?

Tip:

In spherical geometry, all "straight lines" (great circles) intersect, making the concept of parallelism very different from Euclidean geometry!