We continued discussing Book I of Euclid's Elements with our ultimate
goal of proving the Pythagorean Theorem.
Today's goal was to prove (Proposition 32) that the sum of the three
interior angles of a triangle equals two right angles.
In particular we went over the proofs of the following Propositions:
Click on the above links to go directly to the proofs and
commentaries of the results.
In the later Proposition 32, after he invokes the parallel postulate (Postulate 5),
Euclid shows the stronger result that the exterior angle of a triangle equals the
sum of the interior, opposite angles.
Proposition 26 (AAS)
This is the last of Euclid's congruence theorems for triangles.
Euclid's congruence theorems are I.4 (side-angle-side), I.8 (side-side-side),
and this one, I.26 (side and two angles).
Although this is the first proposition about parallel lines, it does not require
the parallel postulate (Postulate 5) as an assumption.
This is the first proposition which depends on the parallel postulate.
As such it does not hold in hyperbolic geometry.