Let us analyze each statement carefully to determine its truth value in the contexts of spherical geometry and Euclidean geometry:

1. In spherical geometry, the sum of the angle measures of a triangle is equal to 180.

   • False. In spherical geometry, the sum of the angles of a triangle is always greater than 180°, because spherical triangles are formed on the surface of a sphere.

2. In both Euclidean and spherical geometry, through any point not on a given line there is a line parallel to the given line.

   • False. This statement reflects the parallel postulate of Euclidean geometry but is not true in spherical geometry. In spherical geometry, there are no parallel lines because all "lines" (great circles) eventually intersect.

3. In both Euclidean and spherical geometry, if all three angles of a triangle have the same measure, then they must all have a measure of 60.

   • False. In Euclidean geometry, this is true for equilateral triangles. However, in spherical geometry, it is possible for an equilateral triangle to have angles greater than 60°.

4. In both Euclidean and spherical geometry, a triangle can have two right angles.

   • True. In Euclidean geometry, this is impossible. However, in spherical geometry, a triangle can have two right angles, as the surface of the sphere allows for such configurations.

5. In spherical geometry, any two given lines must intersect.

   • True. In spherical geometry, "lines" are great circles, and any two great circles on a sphere intersect at two points (antipodal points).

6. None of these are true.

   • False. As noted above, some of the statements are true in specific contexts.

Final Answer:

The true statements are:

• In both Euclidean and spherical geometry, a triangle can have two right angles.
• In spherical geometry, any two given lines must intersect.

Would you like further explanation or visual examples?

Follow-Up Questions:

1. Why is the sum of angles in a spherical triangle always greater than 180°?
2. Can you give examples of configurations where two right angles exist in a spherical triangle?
3. How does the parallel postulate fail in spherical geometry?
4. What is the significance of antipodal points in spherical geometry?
5. How do equilateral triangles differ between Euclidean and spherical geometry?

Tip:

When studying non-Euclidean geometries, always pay attention to how the concepts of "line" and "triangle" differ from their Euclidean definitions.