AAS is not valid in spherical geometry


AAS (http://planetmath.org/AAS) is not valid in spherical geometryMathworldPlanetmath (http://planetmath.org/SphericalGeometry). This fact can be determined as follows:

Let be a line on a sphere and P be one of the two points that is furthest from on the sphere. (It may be beneficial to think of as the equator and P as the .) Let A,B,C such that

  • A, B, and C are distinct;

  • the length of AB¯ is strictly less than the length of AC¯;

  • A, B, and P are not collinearMathworldPlanetmath;

  • A, C, and P are not collinear;

  • B, C, and P are not collinear.

Connect P to each of the three points A, B, and C with line segmentsMathworldPlanetmath. (It may be beneficial to think of these line segments as longitudes.)

PABC

Since is also a circle having P as one of its centers (http://planetmath.org/Center8) with radii AP¯, BP¯, and CP¯, we have that AP¯BP¯CP¯ and that is perpendicularMathworldPlanetmathPlanetmathPlanetmath to each of these line segments. Thus, the triangles ABP and ACP have two pairs of angles congruent and a pair of sides congruent that is not between the congruent angles (actually, two pairs of sides congruent, neither of which is in between the congruent angles). On the other hand, ABP≇ACP because the length of AB¯ is strictly less than the length of AC¯.

Title AAS is not valid in spherical geometry
Canonical name AASIsNotValidInSphericalGeometry
Date of creation 2013-03-22 17:13:00
Last modified on 2013-03-22 17:13:00
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 8
Author Wkbj79 (1863)
Entry type Result
Classification msc 51M10
Synonym SAA is not valid in spherical geometry