SOAR Homework Three
These homework problems are meant to expand your understanding of what
goes on during class. Any you turn in will be graded and returned to
you. Answers may or may not be posted on the web, depending on demand.
Problems
 Let ∠ ABC be a given angle that is not a straight angle.
(That is, the points A, B, and C do not lie on a straight
line.) Show that you may choose a line l such that the lines
AB and BC are the two lines parallel to l (in the sense of
hyperbolic geometry).
(Hint: Look at the bisector of ∠ ABC. The line
l must be perpendicular to this line. Why?)

In the last two classes we've used the fact that the defect
of a triangle is additive. This problem explores this fact. (Here
D(Δ PQR) is the defect of the triangle Δ PQR. This is related to the angle sum S(Δ PQR), which is
simply the sum of the three angles in a triangle. The defect
measures the failure of the angles to sum to what we expect:
D(Δ PQR) = 180^{°}  S(Δ PQR).
 Here is a triangle Δ ABC that has been partitioned
in to two smaller triangles: Δ ACD and Δ BCD (see the pdf file for pictures).
Show that D(Δ ABC) = D(Δ ACD) + D(Δ BCD).

Suppose we can break a triangle Δ ABC into smaller
triangles. Explain why D(Δ ABC) is the sum of the defects
of these smaller triangles. In your explanation, be sure to take
into account at least the following two cases (see the pdf file for pictures)

For each of the following axiom systems, show whether the system is
consistent or not. If it is inconsistent, remove an offending axiom
(or two) so that the remaining axioms are consistent. For any
consistent system, show consistency with a model (and an explanation
of how you created this model).
 In this system, undefined terms are ``bugs,'' ``potatoes'' and
``eats.''
 Axiom 1. There are at least two bugs.
 Axiom 2. There are at least four potatoes.
 Axiom 3. For every pair of potatoes, at least one bug eats both.
 Axiom 4. Every bug eats at least one potato.
 Axiom 5. There is at least one potato that no bug eats.

In this system, the undefined terms are ``boys'' and ``greets.''
 Axiom 1. There are at least three boys.
 Axiom 2. If boy A greets boy B and boy B greets boy C, then boy A
does not greet boy C.
 Axiom 3. Every boy greets himself.
 Axiom 4. Exactly one boy is never greeted by anyone other than
himself.

Make up your own system and prove its consistency or
inconsistency.

For each of the axiom systems in the previous problem, determine if
any of the axioms are independent. Show that your answer is
correct. If, for a particular system, there are no dependent
axioms, introduce one and prove that it is dependent.

Consider the formal axiom system (introduced in class) that had, as
its undefined terms salt, chips, and flavour. The axioms were
 A1 There is at least one salt.
 A2 For any two distinct chips, there is a unique salt that flavours
both of them.
 A3 Each salt flavours at least two chips.
 A4 For each salt, there is at least one chip that it does not
flavour.
 Answer the question posed in class: if there are four chips,
how many salts are there?
 What if there are five chips?

Suppose that the angle of parallelism Π(x) is constant (that is,
suppose that the angle of parallelism is some fixed angle α,
no matter what the distance between the two lines is). Show that
α = 90° and that therefore the geometry under
consideration is Euclidean.
(Hint: Draw a quadrilateral with two opposite sides on
parallel lines and the other two perpendicular to one of the lines.
Where are the angles of parallelism in this picture?)

Consider a circle with an inscribed triangle. The long side of the
triangle is a diameter of the circle. (See the pdf file for the picture.)
 Prove that the angle opposite the diameter is a right angle in
Euclidean geometry.

Prove that the angle opposite the diameter is an acute angle in
hyperbolic geometry.

Here is a proof we were trying to complete in class. We were
proving that, given two parallel lines l and m and a distance
x, one can find a point on m so that the distance from that
point to l is x. The picture for the proof (explained during
the problem) can be seen in the pdf
file.
Here is a sketch of the proof we were going through:
 Choose a point P on m, drop a perpendicular to l and mark
this point Q.

Assuming (for now) that x < PQ, mark a point R on PQ so that
QR = x. Draw the line n through R parallel to l (but
in the opposite direction to m).

Mark the point where m and n intersect as S. Now find T
on m so that TS = RS. Drop a perpendicular from T to
l; mark this point U.
Your job is to complete this proof.
 Complete the above proof. That is, assuming x = QR < PQ, show
that TU = QR (so TU=x, as desired).
(Hint: Show that the triangles Δ STV and
Δ SRV are congruent. Why does ∠ TSV = ∠ RSV?)

Repeat the above prove in the case that x > PQ. This means that
we must extend QP past m so that QR=x can be greater than QP.
These problems are also available as a pdf file.
Solutions
Please email Peter
if you are interested in answers or solutions for the web. Thanks.
SOAR Winter 2003 Course Homepage