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parallelogram theorems
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(Theorem)
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Theorem 1. The opposite sides of a parallelogram are congruent.
Theorem 2. If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.
Theorem 3. If one pair of opposite sides of a quadrilateral are both parallel and congruent, the quadrilateral is a parallelogram.
Theorem 4. The diagonals of a parallelogram bisect each other.
Theorem 5. If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.
All of the above theorems hold in Euclidean geometry, but not in hyperbolic geometry. These theorems do not even make sense in spherical geometry because there are no parallelograms!
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"parallelogram theorems" is owned by pahio. [ full author list (2) ]
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(view preamble)
Cross-references: spherical geometry, hyperbolic geometry, Euclidean geometry, diagonals, parallel, quadrilateral, ASA, side, triangles, alternate interior angles, parallel lines, transversal, line, congruent, parallelogram, opposite sides
There are 3 references to this entry.
This is version 8 of parallelogram theorems, born on 2007-06-14, modified 2007-06-14.
Object id is 9598, canonical name is ParallelogramTheorems.
Accessed 2172 times total.
Classification:
| AMS MSC: | 51-01 (Geometry :: Instructional exposition ) | | | 51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries) |
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Pending Errata and Addenda
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