Math 402, Assignment 7, Fall 2006

Due Wed. March 12 (at the beginning of class)

1. Consider the following statements:
   EEAT: The exterior angle of a triangle is greater than either opposite interior angle.
   PTDI: Two distinct parallel transports don't intersect.
   Prove PTDI for E^2 and H^2 directly without using EEAT or any theorems we proved useing EEAT or PTDI.
   
   HINT: We talked about this in class; look for symmetries.

2. Consider the following statement: Given a triangle ABC, the length of side AC is less than the length of side AB plus the length of side BC.
   1. Prove the statement on E^2 and H^2.

      (Hint: Pick D on ray AB past B so that the segments BD and BC have the same length.)

   2. Give a counter-example to show that this is false on S^2.

3. Prove that the hypotenuse of a right triangle on E^2 or H^2 is longer than either leg.

4. We have discussed a number of triangle congruence theorems: SAS, SSS, ASA, SSA, and AAS.
   1. Which are true for triangles on hyperbolic space?
   2. The proof of all the theorems that you listed above begins more or less the same way -- by lining up congruent parts of the triangle. Then each proof relies on a different key idea. For each theorem, describe this key idea in 20 words or less.
   3. Which theorems are still true for all trilaterals on hyperbolic space? This just means that we include the degenerate trilaterals where all the vertices lie on a line.