Parallels: Assumptions

I've had the intent for some time now to do a little non-Euclidean geometry with my students.

I didn't want it to be mere history. A bit of history is fine, but mostly it should be mathematics. This means that we should prove a set of results.

But what results? I'd like to shock a bit, and so I decided to seek out a proof that, in the variety of non-Euclidean geometry that is called hyperbolic, the sum of the angles of a triangle is less than 180 degrees.

Of course the proof must be elementary. My students are quite bright, but they're only beginners. They have only the resources of that part of Euclidean geometry that we've developed. Thus my task was to find such a proof.

I think I have it. The technique comes from Saccheri. In a set of posts titled Parallels, I'll outline the proof.

Today I'll list those assumptions on which I'll draw. Some are definitions, some are postulates, some are theorems. Which is which is irrelevant. All that matters is that they'll be in place when on that day late in the second semester I begin.

I wish my assumptions to be, as it were, geometry-neutral. I wish them to hold in both Euclidean and hyperbolic, non-Euclidean geometry. Thus I do not include the Parallel Postulate (or any proposition equivalent to it) in the list.

The concept of congruence is key, and so I'll begin there.


Assumptions

1. Vertical angles are congruent
2. In congruent polygons, sides and angles can be paired up in such a way that sides which correspond and angles which correspond have the same measure.
3. In congruent triangles, side which correspond lie opposite angles which correspond.
4. SAS. If two sides and an included angle of one triangle are congruent to two sides and an included angle of a second triangle, then those triangles are congruent.
5. SSS. If the three sides of one triangle are congruent to the three sides of another, then those triangles are congruent.
6. ASA. If two angles and an included side of one triangle are congruent to two angles and an included side of a second triangle, then those triangles are congruent.
7. HL. If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg, respectively, of a second right triangle, then those triangles are congruent.
8. Through a pair of given points a line may be constructed. It is unique.
9. Lines are infinite in extent.

On one variety of non-Euclidean geometry - elliptic - lines are finite in extent. On the other - hyperbolic - lines are infinite, as they are in Euclidean geometry.