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.