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can be phrased as: Let P be a point, I a line, and P not incident with 7. Then there exists one and only
one line m which is incident with P and parallel to 1. If this axiom doesn't hold we could investigate the
worlds in which this is false (no parallel lines, or infinitely many). This parallel property was the basis of this
separation: Euclidean and non-Euclidean. Now, we are discussing Taxicab Geometry and have stated this is a
Non-Euclidean Geometry.
i. The parallel property does hold for Taxicab Geometry. Write some reasons of how you could convince
someone that this is true.
ii. We describe Taxicab Geometry as a non-Euclidean geometry, so something essentially "Euclidean" must
fail in our new Taxicab world. Answer the following questions concerning the two triangles in the images:
B
A
A
C'
A. Does dr(A, B) = dr(A', B')?
B. Does dr(A, C) = dr(A', C')?
C. Does mZBAC = mzB' A'C'? (Measurement of angles. Do not be confused with the talk about in
class.)
D. Do ABC and AA'B'C' have the hypothesis of Congruence Axiom 6 (pg 121 in official textbook),
or Side Angle Side?
E. Is ABCAA'B'C'? Why or why not?
