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Prove that the Hyperbolic Rectangle Does Not Exist

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Using the Universal Hyperbolic Theorem we can show that the sum of the angles of a quadrilateral will be less than 360 degrees in hyperbolic geometry. The formal definition of a rectangle is a quadrilateral with four right angles therefore this figure does not exist in hyperbolic geometry.

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Solution Summary

This solution is comprised of a proof that uses the formal definition of a rectangle and shows that in hyperbolic space a quadrilateral can not have four right angles.

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This fact of the nonexistence of rectangles is sometimes sited as a lemma of the Universal Hyperbolic Theorem. I think is better illustrated by looking at angle sums of triangles and quadrilaterals. If you ...

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