Math 360, Dr. Vaden-Goad, Spring 2006, final exam, name _____________________
Think carefully, show all work
Fold test lengthwise with essay on the outside
Questions 1 through 10 are over the same material as the first test. If your score on these
questions is higher than your score on the first test, they will substitute. In any event,
they will form one third of your final exam grade. For this portion of the test, assume the
incidence postulates are satisfied and give careful, complete proofs.
1. essay as assigned
2. (Carefully and fully) Define affine plane. An affine plane is
3. (Carefully and fully) State any incidence postulate (you do not have to give its number,
just state it):
4. (Carefully and fully) State the hyperbolic parallel property. A model satisfies the
hyperbolic parallel property provided:
5. (Carefully and fully) Define complete. An axiom system is complete provided
6. Use sets to give a model of an incidence geometry with exactly 6 points.
7. Select an incidence postulate and prove that your example in problem 6 satisfies this
postulate.
8. Let S consist of 9 points and let L be the set of all three-member subsets of S. Let
incidence be defined by set membership. Prove or disprove that S and L form an
incidence geometry.
9. Let S = ( x, y) | x  R and y  R be called a set of points and let L consist of subsets
luvw of S with u,v,w real numbers such that at least one of u, v not zero defined by luvw =
( x, y)  S | ux  vy  w. We call L the set of lines. Note that each line is a set of points.
Incidence is set membership. (This is just the regular Cartesian plane and regular lines.)
Prove that the Euclidean Parallel Postulate holds in this model. Use only the definitions
given and facts you know from algebra, but not anything about graphing. Do not use
“same slope means parallel.”
10. Prove, disprove or show independent of the incidence postulates: The projective
completion of an affine plane satisfies the elliptic parallel property.
Questions 11 through 20 are over the same material as the second test. If your score on
these questions is higher than your score on the second test, they will substitute. In any
event, they will form one third of your final exam grade. Proofs in this section may omit
steps based on incidence and betweenness geometry. Any incidence or betweenness
facts evident from the diagrams given may be assumed.
11. essay as assigned
12. (carefully and fully) Define the defect of a triangle. The defect of ABC is
13. (carefully and fully) State the line separation property.
14. (carefully and fully) State the circular continuity principle.
15. (carefully and fully) State the crossbar theorem.
16. Define the interior of an angle. The interior of ABC is
17. Prove, disprove or show independent of neutral geometry: In ABC if
A  B  C then AB  BC  AC .
18. Prove, disprove or show independent of neutral geometry: For any ABC , if
AB  AC , the angle bisector of A is perpendicular to BC.
19. Prove, disprove or show independent of neutral geometry: If the diagonals of a
convex quadrilateral bisect each other, then opposite sides of the quadrilateral are
congruent.
C
B
E
A
D
20. Prove, disprove or show independent of neutral geometry: If P is a point not on line
a, and s and t are lines through P perpendicular to a, then s = t.
Questions 21 through 30 are over the same material as the third test. If your score on
these questions is higher than your score on the third test, they will substitute. In any
event, they will form one third of your final exam grade. For 27 through ec, Assume
Euclidean geometry.
21. essay as assigned
22. In the Beltrami-Klein model, what are the K-lines?
23. If a reflection carries A=(6,3) to A’=(8,5), what is the equation of the line of
reflection?
24. Explain why it is true that if Euclidean Geometry is consistent, so is Non-Euclidean
Geometry. (short answer; don’t just cite the meta-theorem)
25. State a theorem about parallel lines and transversals that is true in Euclidean
Geometry, but false in non-Euclidean geometry.
26. If the circle of radius 5 centered at (2,3) is glide reflected seven units along the y axis,
completely specify its image. (give shape and parameters)
27. Find the formula for a reflection of the coordinate plane about the line y = -x.
28. If the figure is a circle with center C, find the measure of AED
m BAD = 64.80
E
B
C
A
D
29. Prove the fundamental decomposition theorem: Any isometry of the plane may be
expressed as the composition of three or fewer reflections. You may use anything we
have done.
30. In the diagram, Prove that OAOA'  r 2 . You may assume that things that look like
circles, centers, radii and tangents are circles, centers, radii and tangents.
r
O
A
A'