Exterior angle theorem

This is a fundamental result in absolute geometry because its proof does not depend upon the parallel postulate.

This result, which depends upon Euclid's parallel postulate will be referred to as the "High school exterior angle theorem" (HSEAT) to distinguish it from Euclid's exterior angle theorem.

[5][6][7] Euclid proves the exterior angle theorem by: By congruent triangles we can conclude that ∠ BAC = ∠ ECF and ∠ ECF is smaller than ∠ ECD, ∠ ECD = ∠ ACD therefore ∠ BAC is smaller than ∠ ACD and the same can be done for the angle ∠ CBA by bisecting BC.

The other interior angle (at the North Pole) can be made larger than 90°, further emphasizing the failure of this statement.

The HSEAT is logically equivalent to the Euclidean statement that the sum of angles of a triangle is 180°.

Small triangles may behave in a nearly Euclidean manner, but the exterior angles at the base of the large triangle are 90°, a contradiction to the Euclid's exterior angle theorem.
Illustration of proof of the HSEAT