spherical geometry
Spherical geometry is the geometry you actually walk every day: the geometry of the surface of a ball, of which the most familiar example is the Earth. It is not an abstraction invented to spite Euclid — sailors and astronomers needed it for centuries to navigate and to chart the heavens, long before anyone called it 'non-Euclidean'. On a sphere, the role of a 'straight line' — the shortest path between two points, the route a taut string or a great-circle flight follows — is played by a great circle: a circle whose plane passes through the centre of the sphere, like the equator or any meridian.
Once you accept great circles as lines, the rules diverge sharply from the flat plane. There are no parallels: any two great circles intersect (in fact at two opposite points). Triangles, made of three great-circle arcs, have angles that always sum to MORE than 180 degrees; the excess over 180 is called the spherical excess, and it is exactly proportional to the triangle's area (Girard's theorem). A triangle can even have three right angles, summing to 270 degrees. There is also a maximum-size effect: lines are finite (a great circle has length 2 pi R), so you cannot travel forever in a straight line without returning.
Two honest cautions. First, spherical geometry is technically not a fully 'proper' geometry in the strict axiomatic sense, because two great circles meet in TWO points rather than one, so 'two points determine a unique line' fails for antipodal pairs; gluing antipodes together fixes this and yields elliptic geometry. Second, ordinary circles of latitude (except the equator) are NOT lines — they are not great circles, so the shortest air route between two same-latitude cities curves poleward, which is why long flights look bowed on a flat map. Spherical geometry is the everyday face of positive curvature.
On the Earth (radius about 6371 km), the shortest flight from one city to another at the same latitude does NOT follow that latitude line; it bows toward the nearer pole along a great circle. A spherical triangle with one vertex at the North Pole and two on the equator, separated by 90 degrees of longitude, has all three angles equal to 90 degrees, an excess of 270 - 180 = 90 degrees.
Great circles are the lines; no parallels, angle sums exceed 180 degrees, and lines have finite length 2 pi R.
Latitude circles are not 'lines' on a sphere except the equator — only great circles are. And because two great circles meet twice, the pure sphere is not quite elliptic geometry until antipodal points are identified.