# SOAR Homework Four

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.

1. In class I mentioned Bolyai's theorem, which says that if two polygonal regions have the same area, then they are ``equivalent by finite decomposition.'' (This means that there is a single finite set of triangles that can tile each of the two polygons.) Explain why this theorem is true for rectangles in Euclidean geometry.
2. Let Δ ABΩ be an ideal triangle. Show that the sum of ∠A and ∠B is less than 180°.

3. Let Δ ABΩ and Δ A′B′Ω′ be ideal triangles. Assume that ∠A = ∠A′ and ∠B = ∠B′. Show that AB = A′B′. (This can be thought of as an ``Angle-Angle-Angle'' congruence theorem for ideal triangles in hyperbolic space. Remember that we already have such a theorem for regular triangles.)

4. Suppose you are given two ideal triangles Δ ABΩ and Δ A′B′Ω′ with ∠A = ∠A′.
1. Suppose AB > A′B′. What can you say about ∠B compared with ∠B′?

2. Suppose ∠B > ∠B′. What can you say about AB compared with A′B′?

5. In class we saw that there is not necessarily a line through two distinct ultra-ideal points Γ1 and Γ2. Given an ideal point Ω and an ultra-ideal point Γ, is there a line through Ω and Γ? Is it unique? (That is, can there be more than one?)
6. Here is an outline of a proof we skipped in class. We prove that, for any two triangles Δ ABC and Δ A′B′C′, we have
 A(Δ A′B′C′) A(Δ ABC) = D(Δ A′B′C′) D(Δ ABC) .
(*)
In the proof, we'll be using the following fact, often called the comparison theorem:
Suppose x > m/n if and only if y > m/n for any fraction m/n. Then x=y.
Before we start, you should convince yourself that this is true. (The idea is simply that this means x and y are bounded - above and below - by the same rational numbers. If they were different, there would be a rational number between them. Of course, proving this statement is a little bit off-topic, but you should think about it.)

We begin the proof by noting that if D(Δ ABC) = D(Δ A′B′C′), then A(Δ ABC) = A(Δ A′B′C′) by Bolyai's theorem. So we'll assume that D(Δ ABC) > D(Δ A′B′C′) (if they're different, then one of them must be larger).

We choose Q on BC so that D(Δ ABQ) = D(Δ A′B′C′) (to see the picture, please see the pdf file). Space the points P1, P2, and so on so that there are n triangles Δ APkPk+1, each with the same defect, namely D(Δ APk Pk+1) = (1/n) D(Δ ABC). (These points are not presumed to be evenly spaced along BC.) Pick m; we'll show that
 m n < A(Δ A′B′C′) A(Δ ABC) if and only if m n < D(Δ A′B′C′) D(Δ ABC)
which, by the comparison theorem, is enough to prove equation (*).

1. Show that
 m n < A(Δ A′B′C′) A(Δ ABC) if and only if m n A (Δ ABC) < A( Δ ABQ).
2. Show that for any k (and in particular for k=m)
 A(Δ ABPk) < A(Δ ABQ)   if and only if   D(Δ ABPk) < D(Δ ABQ).
3. Explain why (m/n) D(Δ ABC) = D(Δ ABPm) and (m/n) A(Δ ABC) = A(Δ ABPm). Hint: Use Bolyai's theorem which, in this case, you can take to say that if the defects are equal, the areas are equal.
4. Combine the above to show equation (*).
7. In class we saw that it was possible to have an ideal triangle with area πk2 (because the defect was 180° or, equivalently, the angle sum was 0). Anton asked if it was possible to have similar (but non-congruent) such triangles. Answer Anton's question.
Hint: Consider the models. What does such a triangle look like in your favorite model? Is it possible to have two different such triangles with a common corner Ω?

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