Math 402, Assignment 4,

Due Wed. February 13 (at the beginning of class)

1. Your answers to this problem should work on any space. (Remember that the protractor axiom always holds.)
   1. Define the bisector of an angle. (It should be a ray.)
   2. Explain why there exists a unique bisector.
   3. Assume that you can reflect across the angle bisector. Give a (paragraph) proof that this reflection takes one of the rays to the other. (Every proof should include a clear statement of what you are claiming.)
   4. Now go back to your proof and convert it to a two-column proof. Try to fill in every last detail, reducing every claim back to a definition, axiom, or something that we have already proved.
   
    
2. 1. State and prove the Ray Congruence Theorem for E^2, S^2, and H^2.
   2. As in the previous problem, convert this proof to a two-column proof.
   3. For the the first two problems, you did two proofs of the same theorem. In each case, which proof do you like better? Why?
   
    
3. 1. Prove that if a rotation on $E^2$ or $H^2$ is not the identity, then it has only one fixed point.
   2. What happens on S^2? (Be as explicit as possible.)
      Why doesn't your proof from the first part work here?