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Question: we will call a noneuclidean geometry a geometry in which...

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We will call a “non-Euclidean geometry” a geometry in which Eu- clid’s five postulates do not all hold.

  1. (a) Give an example of a non-Euclidean geometry, and explain how you know it is non-Euclidean.

  2. (b) In most (but not all) non-Euclidean geometries, it’s Euclid’s Fifth pos- tulate that fails to hold. Give another statement that is equivalent to Euclid’s fifth postulate. (You may choose any you like - from the text or by looking this up.) Is this statement true in the geometry you discussed in part (a)? Explain why or why not, drawing some pictures to illustrate.

  3. (c) Non-Euclidean geometries can be strange places. Explain why the Pythagorean Theorem (as we usually state it) does not hold in spherical geometry.

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