Axiomatic Geometry - Mathematics 360 - Spring 2009

For Homework 17

1. Prove for every point p in D, there is a transformation T in H such that T(p) is on the x-axis.
2. Prove for every point p in D that's on the x-axis, there is a transformation T in H such that T(p) = 0. Hint: find a hyperbolic Mobius transformation T in H such that T(1)=1, T(-1)=-1, and T(p)=0 by converting to the "w-plane" and back.
3. Prove all points in D are congruent. (Hint: show all points are congruent to 0.)
4. Prove all lines in D are congruent. (Hint: something similar to to last problem's hint.)
5. Can two distinct hyperbolic lines be tangent (i.e. intersect at one point only, with the angle of intersection zero)? Prove your answer.
6. Are any two line segments in hyperbolic geometry congruent? Prove your answer. (A line segment is the "portion" of a hyperbolic line that's between two distinct points on that line.)  Hint: think about Exercise B.