In geometry, a proof is used to present the steps used to arrive at an argument of a mathematical postulate or theorem. An initial claim is presented, and the student is asked to prove it through deductive reasoning, which includes a series of statements linked together to prove the claim. The three basic formats of proofs are two-column proofs, flow proofs and paragraph proofs, which all include statements and reasons. While there is no standard formula for developing and writing a proof, there are strategies for solving this type of math problem.
Basics of Proofs
In its most basic form, a proof includes a list of statements with corresponding reasons giving validity to the statement. Each statement presents an observation from the given information to prove the claim, and each reason uses an already-proven truth, such as a definition or property, as verification for the statement. The logic behind proofs and reasons for proof remain the same regardless of the format.
Defining Geometry Terms
Every proof will begin with a given statement, a fact given as a first step to proving another statement. The given statement may be broken into terms that can be defined. If the given information involves a geometric term that can be defined, make a note of it. For example, if the proof involves an isosceles triangle, noting the definition and characteristics of the special triangle can form a basis for your investigation. Additionally, there may be theorems that are relevant to the proof. Theorems are statements that have been proven through the process of deductive reasoning and can be used to support other claims.
Using Postulates and Diagrams
Postulates are accepted without proof and are statements that can be used as reasons in a proof. Reviewing postulates and identifying their place in a proof is another step in reasoning to prove an initial claim. Examples of common postulates are the reflexive property of equality, the symmetric property of equality and the transitive property of equality. Some proofs will present an initial diagram that may be used for given information. Symbols of equality, right angles and angle measures given in a diagram can be used as statements in a proof.
Constructing the Proof
Once all of the data has been gleaned from the initial claim, given information and any diagrams, constructing the proof begins with stating the hypothesis and the given information. The method of proof should be listed in a series of statements, which are each justified by the given information, a definition, and either a postulate or an already-proven theorem. Finally, the conclusion contains the information that was to be proven with a final justification, providing the last step in deductive reasoning.