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Euclid's parallel postulate, hyperbolic and spherical geometry

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A. Discuss differences between neutral geometry and Euclidean geometry.

B. Explain the importance of Euclid's parallel postulate and how this was important to the development of hyperbolic and spherical geometries.

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https://brainmass.com/math/geometry-and-topology/euclids-parallel-postulate-hyperbolic-spherical-geometry-581305

Solution Preview

A)
Euclid had five postulates:

1) A straight line can be drawn from any point to any point.
2) A finite straight line can be extended infinitely in a straight line.
3) A circle can be drawn given any center and distance.
4) All right angles are equal to one another.
5) If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two straight lines, if extended indefinitely, will meet on that side on which are the angles less than the sum of two right angles. (You may have seen equivalent statements of this postulate, such as Playfair's ...

Solution Summary

The solution discusses Euclid's parallel postulate and its importance to geometry. Additionally it explains the difference between neutral geometry and Euclidean geometry.

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