Proposition 16

This proposition proves that when a side of a triangle is extended, the exterior angle formed is larger than both opposite interior angles. In geometry, the definition of one thing being larger than another is that the smaller thing can "fit inside of" the larger thing, and this is exactly what Euclid does; he comes up with an elaborate construction to fit the opposite interior angles into the exterior angles. This begins with bisecting AC at E, then extending BE into BF so that the length of BE equals the length of EF. It's easy to see that triangle ABE is similar to triangle CFE, meaning the angle A equals the angle ECF. The angle ECF is clearly contained in the exterior angle, therefore the angle A must also be. To prove that the exterior angle is greater than angle B, Euclid extends AC to created another exterior angle equivalent to the original one. From here, the same construction is repeated using BC instead of AC, proving that the angle B can be placed inside the exterior angle.