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Indirect Proof in Geometry: Definition & Examples

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

There are many different methods that can be used to prove a given theory. One of those methods is indirect proof. In this lesson, we will explore indirect proof and learn the steps taken to use this method to prove a given theory.

Direct vs. Indirect

There is always more than one way to get from point A to point B. Imagine you and I are out for a walk. We are starting at point A and ending at point B. One way to go about this is along a straight path from point A to point B, and the other is a different path that has more twists and turns.

The second path isn't anymore difficult; it's just different than the straight path. Normally, the straight path is the path most often followed, because it is a direct shot from one point to the other. The path that is a bit different is the path less traveled, because it is an indirect way of getting from point A to point B.

Direct vs. Indirect
indirect proof 1

These paths are a great analogy for direct and indirect proofs in geometry. A direct proof in geometry is the most common form of proof used. In a direct proof, we logically deduce the conclusion directly using facts pertaining to the specific situation. This can be thought of as a direct shot, or a straight path to get from an assumption to a conclusion.

A less common form of proof in geometry, though equally effective, is the indirect proof. In an indirect proof, we go about proving a conclusion in a roundabout way. This can be thought of as the path less traveled. Both paths can be used to get to the same place and both are equally effective. They are just two different ways of getting the same result, where the direct proof is more straightforward than the indirect proof. In this lesson, we are going to concentrate on indirect proofs.

Indirect Proof

When we use an indirect proof to prove a theory, we follow three steps.

1.) Start by assuming that the theory is false

2.) Next, we go about our proof and eventually run into a contradiction, that is something that doesn't make sense.

3.) The contradiction from step 2 proves our assumption of the theory being false not to be the case, so the theory must be true.

An indirect proof is also called a proof by contradiction, because we are literally looking for a contradiction to a theory being false in order to prove that the theory is true.

Indirect Proof
Indirect Proof 2

When to Use Indirect Proof

A good way to determine if you should use the indirect proof method is to ask yourself the following question : What if this wasn't true?

Indirect Proof
Indirect Proof 3

If we ask ourselves this question, and the answer is that a contradiction would happen, then using the indirect proof method is a good idea.

For example, imagine you are taking a certain type of medication, and you can't remember if you remembered to take it on a certain day. You ask yourself the question what if I didn't take it today? You realize that you had 10 pills left yesterday. If you didn't take it there would still be 10 pills left today, but you counted the pills earlier and there are only 9 left. Therefore, it wouldn't make any sense that you didn't take it today, so you must have taken your pill today. You've just proven that you took your pill using an indirect proof.

Example

Let's look at an example of an indirect proof in geometry. Consider the image below.

Example
Indirect Proof 4

In this image, line segment DB is perpendicular to line segment AC. We are going to use an indirect proof to prove that angle ABD is not 180 degrees.

Proof:

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