### MATH325: Sample Final

Name:

You may use pencil, pens, protractor, ruler, compass, scissors, calculator, book or notes on this test. You may not use a cell phone or any device with the ability to communicate outside the room. Please take your time on each problem and show your work in the space provided. If you get stuck on a problem, move on to the next and return when you have time.

1. (30 pts) Using the techniques demonstrated in Chapter 1 of the text, prove the Angle-Side-Angle theorem. In other words, prove that if two triangles ABC and DEF have ∠ABC ≡ ∠DEF, ∠BCA ≡ ∠EFD and |BC| = |EF|, then triangle ABC is congruent to triangle DEF.)
2. (20 pts, based on Exercise 3.6) List the steps needed to construct a regular octagon (8-gon) using a straightedge and compass.

3. (20 pts) Consider Corollary 1.7.4 on page 27 of the text book. Is this theorem true in Hyperbolic geometry? Why or why not?

4. In taxicab geometry, define a rhombus to be a collection of four points (intersections) A, B, C, D for which |AB| = |BC| = |CD| = |DA|.

a) (10 pts) Given the points A and B below find two points C and D so that ABCD is a rhombus, or explain why that is not possible.
b) (20 pts) Is it true that if ABCD is a rhombus there is only one other rhombus in the taxicab plane that shares edge AB? Justify your answer. (In other words, can there be more than one rhombus on the "same side" of edge AB?)

Tentative Solutions

Outcomes:

• Know Euclid's five axioms and the implications of accepting, changing or omitting an axiom.
• Prove theorems based on a given set of axioms, lemmas, and previously proved theorems.