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 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 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.
The Launching Pad
Sometimes you meet with a problem that asks you to prove something is not true. In such a case, you can use the launching pad to blast yourself away from having to directly deal with the problem, instead providing a counterexample to show how something’s not true. When you use a counterexample, you only need one good counterexample to prove your point, and the proof will be valid. For example, if you need to validate or invalidate the statement “All trapezoids are parallelograms,” you only need to provide one example of a trapezoid that is not a parallelogram. You could do this by drawing a trapezoid with only two parallel sides. The existence of the shape you just drew would disprove the statement “All trapezoids are parallelograms.”
Just as geometry is a visual mathematics, the flowchart, or flow proof, is a visual type of proof. In a flow proof, you begin by writing down or drawing all the information you know next to one another. From here, make inferences, writing them on the line below. In doing this, you are “stacking” your information, making something like an upside-down pyramid. You use the information you have to make more inferences on the lines below until you get to the bottom, a single statement that proves the problem. For example, you might have a line L that crosses through point P of the line MN, and the question asks you to prove MP = PN given that L bisects MN. You could start by writing the given information, writing “L bisects MN at P” at the top. Below it, write the information that follows from the given information: Bisections produce two congruent segments of a line. Next to this statement, write a geometric fact that will help you get to the proof; for this problem, the fact that congruent line segments are equal in length helps. Write that. Below these two pieces of information, you can write the conclusion, which naturally follows: MP = PN.
About the Author
Having obtained a Master of Science in psychology in East Asia, Damon Verial has been applying his knowledge to related topics since 2010. Having written professionally since 2001, he has been featured in financial publications such as SafeHaven and the McMillian Portfolio. He also runs a financial newsletter at Stock Barometer.