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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Similarity and Differences between adding rational expressions and polynomial expressions.

Difference

When polynomials expressions are added, the like terms are evident by mere inspection as the variables need not be further simplified. But with radical expressions, it is a must that the expressions are simplified to figure the like terms.

Simplification is mandatory while adding radical expressions but while adding polynomial expressions; there is not need for further simplification before addition.

In the above example of addition of polynomial expressions, no further simplification was needed prior to group and addition of like terms.

In the above example with adding rational expressions, simplification is a must before proceeding with categorizing like terms and adding them.

Similarity

The similarity between addition of radical expressions and that of addition of polynomial expressions is that same rules are applied for addition in either case, but in radical expressions rules are applied after simplification.

Radical expression area grouped in similar way as terms in polynomial expressions while adding.

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