Which of the following statements from Euclidean geometry is also true in spherical geometry?

Question

1. A segment is the shortest path between two points.
2. Perpendicular lines intersect at one point.
3. Perpendicular lines form four right angles.
4. A line has infinite length.

Answer 1

In spherical geometry, lines are finite in length, intersecting lines intersect at two points. That rules out 2 and 4.

(3) Perpendicular lines form eight right angles in spherical geometry -- four at each of two intersection points.

(1) The shortest distance between two points in spherical geometry is an arc of a great circle called a geodesic.

It's a tricky question. If one considers only a single intersection point, one might argue for (3). And if one considers a geodesic a "segment" along the surface of the sphere, one might argue for (1). But I think both are technically incorrect -- i.e., none of the four statements is really true of spherical geometry.

Answer 2

A straight line in the Euclidean geometry has as analogue a great circle in the spherical geometry, so in latter case we'll interpret 'line' (the term You have used in Your question) as a great circle and 'segment' as an arc of a great circle. This is a standard assumption in the spherical geometry.

1. If the points are not ends of a diameter of the sphere the shortest distance is precisely that segment of the line between them, whose angular measure is less than π(pi); otherwise there are infinite shortest paths half of a great circle long. To illustrate that, the shortest path between London and the North Pole is the segment of the Greenwich meridian in the Northern hemisphere and not the complementary arc, running southwards from London along the meridian to the South Pole and then northwards to the North Pole along the 180th meridian. The shortest path between the Poles is any meridian.
2. They intersect in 2 points, take the Equator and the great circle, formed by the Greenwich and 180th meridians on the globe.
3. The same example shows that perpendicular lines form 8 right angles.
4. Any line has finite length - the length of a great circle.

Answer 3

It is definitely not b, because they would intersect at two points. It can't be d, because a "line" is a great circle in spherical geometry. So, you have a) that is left. It doesn't say straight line or straight segment. The segment is always the shortest path between two points regardless of the geometry. That's just the definition of a segment.