Spherical Geometry

Prove the statement” there exists a triangle with a sum of angles greater than 180 degrees" is true in spherical geometry.

Let us take Reimann geometry or Spherical geometry.Let a triangle be drawn on a sphere’s surface. All the lines can be made ‘straight’ as all the angles are greater than 180.

Let both the poles be marked on the sphere that is North pole and South pole. Let four points be marked equidistantly around the equator. Let the North pole be joined to the two adjacent points on the Equator and then if these two points are joined together to each other, there are in fact three straight lines on the sphere but each of the angles if 90 degrees. And the sum of the angles of this triangle formed is 270 degrees.

Again there are two types of spherical triangles, one is right spherical triangle which has 1 angle compulsorily of 90 degrees. This type of triangle has more chances to have the sum of all angles(interior) to be less than 180 degrees.The other type of right triangle is oblique spherical triangle where the interiors angles sum is greater than 180 degrees.

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