### 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.

- (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.)
- (20 pts, based on Exercise 3.6) List the steps needed to
construct a regular octagon (8-gon) using a straightedge and compass.

- (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?

- 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.