What Are Proofs in Geometry?
Geometry proofs are used to prove a mathematical concept is true. These proofs use deductive reasoning. Several types of proofs are used. Some of them are two-column proofs, paragraph proofs, and flowchart proofs. Flowchart proofs are particularly useful for teaching beginner geometry concepts. They are a great tool to show students the steps of a proof, and they are self-explanatory. A paragraph proof is a type of geometric proof that includes statements and reasons. However, this type of proof is not usually written out in columns, as the steps are not listed in a chronological order.
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Paragraph proofs are more common for college educators. In a paragraph proof, the statement and reasons are explained in detail. Each step is backed up by a definition or a mathematical reason. It is important to write out all of the steps so that each one supports the next. This is a bit harder for beginners to do.
Two-column proofs are the most common of all geometric proofs. The first column of the two-column proof lists the statements. Another column uses deductive reasoning to provide justifications for each statement. For example, if the two right triangles are ADB and CDB, then both triangles are right triangles. But there are other criteria for proving that a triangle is congruent.
One common example is the parallel postulate. This is a statement that is assumed to be true based on basic geometric principles. If there are two lines in space that do not intersect, they are parallel. Similarly, two circles with the same radius have the same circumference. Both of these statements are based on the same facts. Thus, the proofs are applicable to other claims.
A flowchart proof is a type of geometric proof that demonstrates the steps of a proof by using boxes and arrows. There are also some ways to organize these reasons. Typically, a flowchart proof will begin with the Statement, then show the progression of the ideas through boxes and arrows. After showing the progression, the proof will end with a final conclusion. Unlike other types of proofs, a flowchart proof does not require students to write out every step.
Paragraph proofs are similar to two-column proofs, but they are less organized and may be difficult for beginners to understand. They are primarily used at the collegiate level of geometry. Although they are more complicated to read and follow, they are helpful for students to learn the process of a proof.
As with all types of proofs, there are different styles. Direct proofs are a popular format. Using this style, the goal of the proof is to convince others that the statement is true. Often, a student must provide counter-examples to explain why their answer is not correct. Alternatively, students are given a conjecture and must state whether the conjecture is true or not.
Another style of proof is the biconditional format. Typically, a biconditional format is used when the same line of reasoning can be used to prove several different theorems. An example of this is the HL criteria, which requires a congruent hypotenuse pair and a congruent leg pair. Regardless of which format is used, proofs should have a final line of text stating that the conclusion is correct.