# How to Explain Different Types of Proofs in Geometry ••• Comstock/Stockbyte/Getty Images

Face it: Proofs are not easy. And in geometry, things seem to get worse, as now you have to turn pictures into logical statements, making conclusions based on simple drawings. The different types of proofs you learn in school can be overwhelming at first. But once you understand each type, you’ll find it much easier to wrap your head around when and why to use different types of proofs in geometry.

## The Arrow

The direct proof works like an arrow. You start with the information given and build on it, moving in the direction of the hypothesis you wish to prove. In using the direct proof, you employ inferences, rules from geometry, definitions of geometric shapes and mathematical logic. The direct proof is the most standard type of proof and, for many students, the go-to proof style for solving a geometric problem. For example, if you know that point C is the midpoint of the line AB, you can prove that AC = CB by using the definition of the midpoint: The point that falls equal distance from each end of the line segment. This is working off the definition of the midpoint and counts as a direct proof.

## The Boomerang

The indirect proof is like a boomerang; it allows you to reverse the problem. Instead of working just off the statements and shapes you are given, you change the problem by taking the statement you wish to prove and assuming it’s not true. From there, you show that it cannot possibly not be true, which is enough to prove it is true. Though it sounds confusing, it can simplify many proofs that seem difficult to prove through a direct proof. For example, imagine you have a horizontal line AC that passes through point B, and at point B is a line perpendicular to AC with endpoint D, called line BD. If you want to prove that the measure of angle ABD is 90 degrees, you can start by considering what it would mean if the measure of ABD were not 90 degrees. This would lead you to two impossible conclusions: AC and BD are not perpendicular and AC is not a line. But both of these were facts stated in the problem, which is contradictory. This is enough to prove that ABD is 90 degrees.