Proof by Contradiction

When we are presented with a proposition that we must prove, we can do so through contradiction.
What does this mean?
Until now, we have proven propositions directly, according to the following pattern: because this and that happen... this and that occur... and in this way, the proposition is proven.
For example, if the diagonals of a parallelogram are equal, then the parallelogram is a rectangle.
This is how we proved all the propositions we needed.
Proof by contradiction is a demonstration that is done indirectly.
We prove that it is impossible for the proposition to be false and, therefore, it must be true.
In fact, we do not prove why it is true, but why it cannot be false - this is the proof by contradiction.
Does it make you dizzy?
Let's organize our minds and see the correct order of operations to carry out proofs by contradiction:

1. When we are presented with a proposition, we must remember that there are two options:
   The proposition can be true or false.
2. Let's imagine that the proposition is false and see what would happen.
3. Now let's see the result when the proposition is false. This result should contradict some fact or data.
4. We will clarify that, since a contradiction of facts or data is created when the proposition is false, it must be true, therefore, the option that the proposition is false is excluded.
5. After excluding the option that the proposition is false, we are left with only the other option, that it is true.

Proof by contradiction - the proposition must be true because if it were false, it would lead to an absurdity or contradiction of facts or data.

The proposition goes as follows:
If there are two alternate angles between lines and these angles are equal, then the lines are parallel.
Let's proceed step by step, according to the order of operations for the proof by contradiction:

Let's remember that there are two options: The proposition is true or the proposition is false.

Let's review what would happen if the proposition were false:

We will ask ourselves, is it possible that $∢1=∢2$ (the alternate angles in the illustration are equal), but the lines $A$ and $B$ are not parallel but intersecting?

If the lines are not parallel, that is they are intersecting, it means they will meet at some point and a triangle will be formed:
Let's try to draw a sketch:

Let's see what result came out and where it contradicts some fact.
What is in the triangle that was created?
We have a triangle with an external angle – angle 2
equal to the internal angle - angle 1
that are not adjacent.
This result contradicts the fact that there cannot be an external angle of a triangle that is equal to the internal angle that is not adjacent to it in the same triangle.

Since we have seen what happens when the proposition is false (it contradicts a fact), we can determine that the option of the proposition being false is discarded, therefore, it must be true.
That is, it is not possible for the lines not to be parallel.
The lines must be parallel and never intersect.

We are left only with the option that the given proposition - if there are two alternate angles between lines and these angles are equal, then the lines are parallel - is true.

Great!
Now you also know how to prove propositions by contradiction.
This method will greatly help you solve various types of exercises.