Solutions to Test 1, Math 302, and scoring, W7

Instructions: 
If you wish to have anything on your test reconsidered,
(1) Do NOT write anything on the test papers themselves.
(2) Write your request on a separate piece of paper.
(3) Submit your test, the requests AND your journal.

To compute your relative (curved) score y normalized to a  
mean of 80 and standard deviation of 15 compute  this
           y = (x - 59)*5/7 + 80. 
A normal distribution of 80+/-15 puts 25% in 90-100, 25%
in 80-90, 25% in 70-80, and 25% below 70. Note that as
absolute scores change due to grading corrections, and as
students drop the course, the relative grade may change.

1(a) What are the Lines and Points in the Klein-Beltrami model of
hyperbolic (non-Euclidean) geometry.

The Points are the points inside a disk, the Lines are the open
chords of the disk. Open means that the endpoints of the chords 
are not on that Line.


1(b) Define what it means for one Line to be Perpendicular to another
Line in the Klein-Beltrami model.

If the Line lies on a proper chord (not a diameter) then the 
Perpendicular Lines lie on chords, which when extended pass through
the polar of the chord. The polar of a chord is where the two lines
tangent to the circle at the ends of the chord meet outside the
circel. (6pts) When the Line lies on a diameter, all chords 
Euclidean-perpendicular to this diameter are Perdendicular Lines. 
(5 pts). This problem requires pictures.


1(c) Demonstrate by a construction that if one Line is Perpendicular
to a second Line, then the second is also Perpendicular to the first.

The picture needs to be accurate. That is why I provides two circles
with their centers on the test sheet, along with a file card.


2(a) State the Exterior Angle theorem (Euclid's Proposition 16). 

Draw a figure. For every triangle, an exterior angle is strictly
larger than either of the opposite interior angles.

2(b) Use it to prove that if a line \ell crosses two lines, m and n,
so that opposite interior angles are equal, then m || n. Hint: Vertical
angles are equal. 

The only way to prove this theorem without using the parallel postulate
is by means of 2(a). And for this, you show that, assuming the contrary,
there is a contradiction to the Exterior Angle Theorem. See notes and
class. A picture depicting two parallel lines crossed by a third cannot
be adequate here. This picture assumes what you're proving. A picture
with two non-parallel line crossed by a third suggests you knew the
answer to this question. 

2(c) Prove the Exterior Angle theorem using SAS. Hint: Extend a median
an equal distance beyond the side of a triangle and connect the end
point to another vertex of the original triangle. Draw a picture.

See notes and class. For the remote opposite interior angle only, 
there were 6pts. For the proximate opposite interior angle, you
need the same construction on the other side, AND apply the 
vertical angle theorem. 

3(a) What does it mean for one axiom of an axiom system to be 
independent of the other axioms? And how can you check for
independence? 

An axiom is independent from a set of axioms if it is not a 
consequence of them. (6pts) To check independence, demonstrate models
for two axioms systems, one including the axiom in question,
the other including its negation. (5pts)

4(b) Can you give an example of independence.

If you cited Euclide 5, you also needed to explain why it is 
independent: Because there two different models, one for 
Euclid and one for hyperbolic geometry.

If you cited one of the finite geometries, you need two figures
representing the two models for full credit.

4(c) Write one paragraph (about half a page) that answers the question:
"What is Euclidean geometry?"

What is most important in such questions as this one is NOT to write
sentences that anyone might have written who knows nothing about 
this course. For example: The geometry as invented by Euclid.

Lists of attributes of Euclidean geometry (e.g. "Euclidean geometry is
very important.") should never appear in a short summary.
You don't answer the question "What is the University of Illinois" by
saying "They have a loosing football team." More informative, but 
still not optimal would be "They have two Nobel and one Crafoord 
Prize winners in one year."