let me give you an example of non-Euclidean geometry. Everybody knows a triangle has 180 degrees. Right?
NO! False! That only works on planes not spheres.
For example, assuming the Earth is a perfect sphere, draw a line from the center of the north pole to the equator. That line now forms a 90 degree angle with the equator. Now draw another line from the center of the north pole to the equator, but at a 90 degree angle to the first line. This line is also at a 90 degree angle with the equator. Now connect the 2 lines with another line drawn on the equator.
As you can see, there are three 90 degree angles in this triangle. And 3*90 is 270 degrees; way above 180.
Once we’re going there, let’s also give this example:
Imagine the world is an imperfect sphere, and draw a line from Lakewood to Tel Aviv, and then to the Mir, and then to the best cholent place on thursday night in Beis Yisroel, and then back to Lakewood. As is obvious, there are way more than 180 degrees!
Because neither one is a triangle. Triangle’s have three straight lines; not squiggles.