Hyperbolic geometry, with this definition of distance, turns out to have all the properties of Euclidean geometry apart from those which need the parallel postulate. We can construct triangles and other plane figures of different shapes and sizes, and we can move them around ‘rigidly’ (keeping their hyperbolic shapes and sizes from changing) with as much freedom as we can in Euclidean geometry, so that a natural notion of when two shapes are ‘congruent’ arises, just as in Euclidean geometry, where ‘congruent’ means ‘can be moved around rigidly until they come into coincidence’. All the white
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