It is doubtful that you will ever have a use for any type of non-Euclidean geometry. That being the case, why is a description of it included in "Math & You," which focuses on mathematics that is used in everyday life?
Here is the answer. The author of the book, who happens to be the same person who is writing this commentary, wanted to give you an example of two perfectly good logical systems that disagree with each other, and yet are perfect for modeling their own worlds.
Euclidean Geometry: Is a perfect model for people who live in a plane or in normal 3-dimensional space.
Spherical Geometry: Is a perfect model for people who live on the surface of a sphere.
These two geometries are inconsistent with each other. Both can't be "true" at the same time in the same place, but being "true" isn't really the point.
The Earth is not considered a perfect sphere, but rather is called an "oblate spheroid" with a slight bulge at the equator and flattening at the north and south poles.
To explore the world, moon, and sky in 3D, visit Google Earth.
In spherical geometry, it is also true that a triangle is determined by 3 intersecting lines (great circles).
One such triangle is pictured below.
The question is, is the sum of the 3 angle measures less than 180 degrees, equal to 180 degrees, or greater than 180 degrees? You might try conducting an experiment to measure the 3 angles. To do this, you must be sure that each of the 3 sides is a great circle. You also need to take very careful measurements of the angles.
If you perform this experiment, you will discover that the sum of the angles measures greater than 180 degrees.
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