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JIMMY
Posts:
7
Registered:
12/6/04


Valid Definitions in Spherical Geometry
Posted:
Jun 1, 2000 4:13 PM


How logical do you find the following definitions in spherical geometry??
Similar polygons = all corresponding angles are congruent and lengths of corresponding sides are proportional
Parallelogram = quadrilateral whose opposite sides are parallel
Rectangle = parallelogram whose angles are right angles
These definitions are really illogical in spherical geometry. Here are more logical definitions in spherical geometry:
Similar polygons = all measures of corresponding angles are proportional and all lengths of corresponding sides are proportional. In other words, a spherical triangle whose angles measure 90 90 90 is similar to one whose angles measure 120 120 120 but not to one whose angles measure 90 90 120. This is more logical because if we defined similar polygons the same way they are defined in Euclidean geometry, "similar" and "congruent" would mean the same thing in spherical geometry.
Parallelogram = quadrilateral whose opposite angles are congruent. In other words, a spherical quadrilateral whose angles are 90 120 90 120 is a parallelogram but not one whose angles are 90 120 90 90. This is more logical because if we defined a parallelogram in spherical geometry as a quadrilateral whose opposite sides are parallel, then there would be no parallelograms in spherical geometry.
Rectangle = parallelogram whose consecutive angles are congruent. In other words, a spherical parallelogram whose angles are 120 120 120 120 is a rectangle but not one whose angles are 90 120 90 120. This is more logical because if we defined a rectangle in spherical geometry as a parallelogram whose angles are right angles but defined a parallelogram as stated above, there would be no rectangles in spherical geometry.
Any other terms that need to be defined differently in spherical geometry??



