What Is An Indirect Proof

Indirect proofs are sort of a weird uncle of regular proofs. With an indirect proof, instead of proving that something must exist true, you lot prove information technology indirectly past showing that it tin not be imitation.

Note the not. When your task in a proof is to prove that things are not congruent, not perpendicular, and so on, it's a expressionless giveaway that you lot're dealing with an indirect proof.

For the most part, an indirect proof is very like to a regular proof. What makes information technology unlike is the style it begins and ends. And except for the beginning and finish, to solve an indirect proof, yous use the same techniques and theorems that you would use on regular proofs.

The best manner to explain indirect proofs is by showing you an example. Here you go.

Note 2 peculiar things nigh this odd duck of a proof: the not-congruent symbols in the givens and the testify argument. The one in the prove argument is sort of what makes this an indirect proof.

Here's a game plan showing how you tin tackle this indirect proof. You assume that the prove statement is fake, namely that segment PS is coinciding to segment RS, and then your goal is to arrive at a contradiction of some known true matter (ordinarily a given fact about things that are not congruent, not perpendicular, and and so on). In this trouble, your goal is to show that angle PQS is congruent to angle RQS, which contradicts the given.

One last matter before showing you lot the solution — you lot can write out indirect proofs in the regular format, just many geometry textbooks and teachers present indirect proofs in paragraph class, like this:

Assume the contrary of the prove statement, treating this opposite statement as a given.
Work through the problem as usual, trying to prove the opposite of one of the givens (usually the one that states something is not perpendicular, congruent, or the like).
End by stating that you've reached a contradiction and that, therefore, the prove statement must be truthful.

(Quod E rat D emonstrandum — "which was to be demonstrated" — for all you Latin-speakers out in that location; the residual of you tin just say, "I'm washed!")