Proposition 19

This proposition is a converse of the previous one, stating that if one side is larger than another, then its opposite angle is larger than the other's opposite angle. The proof of this is actually very simple, but leverages a very powerful tool. This tool states the equivalence between the statements "A implies B" and "not B implies not A". The way that you would prove that "A implies B" implies "not B implies not A" is by contradiction. Assume that not-B implies A, and using the fact that A implies B, this means not-B implies B, a contradiction. To prove that "not B implies not A" implies "A implies B", you would again use contradiction. Assume A implies not-B, and since not-B implies not-A, this means A implies not-A, a contradiction.

​

To leverage this tool, we can set A to be "the blue side is larger than the green side", set B to be "the blue angle is larger than the green angle", set not-A to be "the green side is larger than the blue side", and set not-B to be "the green angle is larger than the blue angle". The previous proposition proves that A implies B, so we can use this tool and the proof is already complete. This method is called a proof by contrapositive, and it's really just a special case of proof by contradiction.