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1. What is the nature of the sum of the measures of the angles of a triangle in plane and spherical geometry?
The sum of the measures of the angles in a triangle always add to \(180^{\circ}\) in plane geometry. In spherical geometry, they add to more than \(180^{\circ}\). In fact, you can draw a triangle that has three right angles on a sphere. If you did, it would be exactly one eighth of the sphere (or one quarter of a hemisphere).
2. Is the Pythagorean Theorem valid in spherical geometry?
No. If you can draw a triangle with three right angles, then any two of the sides would be legs and the third side would be the hypotenuse. So, the sum of the squares of the legs could not possibly equal the square of the hypotenuse.
3. What is the smallest sided figure on a plane and on a sphere?
On a plane, a triangle (three sides) is the smallest sided figure. On a sphere however, you can construct a biangle, called a Lune, a two sided figure. A Lune is the intersection of two great circles.
The sum of the measures of the angles in a triangle always add to \(180^{\circ}\) in plane geometry. In spherical geometry, they add to more than \(180^{\circ}\). In fact, you can draw a triangle that has three right angles on a sphere. If you did, it would be exactly one eighth of the sphere (or one quarter of a hemisphere).
2. Is the Pythagorean Theorem valid in spherical geometry?
No. If you can draw a triangle with three right angles, then any two of the sides would be legs and the third side would be the hypotenuse. So, the sum of the squares of the legs could not possibly equal the square of the hypotenuse.
3. What is the smallest sided figure on a plane and on a sphere?
On a plane, a triangle (three sides) is the smallest sided figure. On a sphere however, you can construct a biangle, called a Lune, a two sided figure. A Lune is the intersection of two great circles.