Date: Dec 11, 2007 2:34 PM
Author: Rich Kleinschmidt
Subject: Exterior Angle without parallels?

Hello friends.  Is anyone still out there?

I have been using Harold Jacobs "Geometry-Seeing, Doing, Understanding" for the first time this year.

I have enjoyed his approach and he has given me some good insights for other geometry classes I teach, but....

In Chapter 5 (Theorem 12) he offers the theorem:
An exterior angle of a triangle is greater than either remote interior angle.

Certainly not controversial except that he does it before introducing the parallel postulate. Later he uses it in the NonEuclidean chapter (Theorem 89) to prove:
"If the legs of a birectangular quadrilateral are unequal, the summit angles opposite them are unequal in the same order."

My problem is that I don't think the theorem applies in at least some geometries. For example in spherical, couldn't we have a triangle with the longitude 0, longitude 100 and the equator? Wouldn't the exterior angles be less than or equal the remote interior angles?

Am I missing something or did Harold Jacobs miss something. I think his mistake (P191 step 10) is an assumption about betweeness.

Any insights welcome. (I really like this text for our lower level classes.)

Rich Kleinschmidt
Bishop Brady High School
Concord, NH 03301