Lecture 14 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: Proposition 16 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). Proposition 27 Although this is the first proposition about parallel lines, it does not require the parallel postulate (Postulate 5) as an assumption. Proposition 29 This is the first proposition which depends on the parallel postulate. As such it does not hold in hyperbolic geometry. Proposition 32 Click on the above links to go directly to the proofs and commentaries of the results.