Across
- 5. Instead of assuming a flat space, they explored ________ surfaces.
- 8. In __________ geometry the sum of the angles is exactly 180 degrees.
- 11. ____________geometries emerged in the 1800s.
- 12. Euclidean, spherical, and hyperbolic each have unique rules, especially about ____ Lines
Down
- 1. In __________ geometry triangle angle sums here are less than 180 degrees.
- 2. In _______ geometry parallel lines don’t exist, and triangle angle sums are greater than 180 degrees.
- 3. Euclidean, spherical, and hyperbolic each have unique rules,about Triangle ____ as well.
- 4. Mathematicians began questioning _____’s postulates.
- 6. __________(Last Name Only) was a German mathematician who developed spherical geometry, also known as elliptic geometry. He questioned the need for Euclid’s parallel postulate and explored what happened on a positively curved surface. His work later helped Einstein describe gravity and spacetime in general relativity.Nikolai Lobachevsky ____________(Last Name Only), a Russian mathematician, developed hyperbolic geometry. He proposed that through a point not on a line, you could draw infinite non-intersecting lines. His work was revolutionary—it was the first published system that truly broke from Euclidean thinking.
- 7. Do parallel lines meet in Hyperbolic geometry: They don’t meet, but there are _______ lines that stay “parallel” through a given point.
- 9. Do parallel lines meet in Spherical geometry: _______—they eventually intersect.
- 10. Do parallel lines meet in Euclidean geometry: _______.