Question

Angle bisectors.

a. Define angle bisector.

b. claim If you can reflect across the bisector of an angle, the reflection exchanges the two rays.) Give a careful and detailed two column proof of this claim. (This means that you put the steps on the left and the reason for each step on the right.)

c. Make sure that at every step you justify your claim in terms of a definition or an axiom.

d. Which (parts of) axioms did you use? What spaces does this proof work on?

====================================================================================================An axiom means "A proposition that commends itself to general acceptance; a well established or universally-conceded principle..." (OED2). Often, one assumes the following statements are true. However, they are not true on every space. Therefore, we will check if each statement is true on each space.

1. The "Incidence Axiom":There is at least one straight line between two (distinct) points.
2. There is at most one straight line between two (distinct) points.
3. The "Ruler Axiom":You can travel any distance along a straight line in either direction.
4. If you travel along any straight line, you will never go over the same point more than once.
5. The "Plane Separation Axiom":The complement of a straight line (what you have left when you cut along it) consists of two pieces so that:If two points lie in difference pieces, the line segment between them crosses the line.
6. If two points lie in the same piece, the line segment between them does not cross the line
7. The "Protractor Axiom":There is at least one straight line through any point in any direction.
8. There is at most one straight line through any point in any direction.
9. The "Mirror Axiom": There is a global reflection through every straight line.