How to Do Proofs in Geometry

Shapes have specific geometric properties.

High school and college students learn about geometric proofs, a set of reasons that prove a statement about a geometric relationship. Many professions make use of the ability to prove a geometric relationship. Architects, fashion designers, math teachers and video game developers often develop proofs to excel in their profession.

The successful completion of a geometric proof requires that you select a method of proof and devise a logical plan to prove the required statement. Although a proof can be difficult, the task will become much easier if you first examine completed geometric proofs, known as theorems.

Things You'll Need

• Geometry textbook
• Book of geometric proofs
• Paper
• Pencil

Instructions

1. Identify what needs to be proved. If it's a geometry proof problem, the statement to be proved will often be clearly stated. From that statement work backward. Identify what has to be proved so that a statement of proof can be made about the final statement. For each of these prerequisite proofs, determine which geometric definitions, postulates or properties will establish the prerequisite proofs as true. Examine the givens that the problem provides. Pinpoint which givens prove any of the needed prerequisite proofs to prove the final statement.

2. Determine the best method of proof from the different methods of proof. These include the direct proof and the indirect proof. Consider using the direct proof if the final statement can be deduced from given or known statements, or other reasons. Use the indirect proof only as a last resort. Remember that an indirect proof attempts to prove by showing that all other possible cases are not true.

3. Determine the best method to explain the proof. Select between the two-column proof, the paragraph proof and flow chart proof. For formal teaching and geometry tests, the two-column proof is often required. Understand that the two-column proof has one column for reasons and another column for statements. Select a paragraph proof if it is easier to explain the proof with words rather than sequential geometric statements of reason. Remember that geometric symbols used in statements of reason sometimes are difficult to follow. Select a flow chart proof if you want to emphasize the concepts of the proof. Flow chart proofs, which are a systematic sequence of geometric figures that prove, often are easier for beginners to understand.

4. Identify the reasons for each statement about the proof. Remember that in a geometric proof reasons include given information, geometric postulates, geometric definitions, geometric theorems and geometric properties. Establish a clear understanding of these reasons and what true statements they imply. From a geometry textbook, make a list of all the geometric postulates, definitions, theorems and properties. Remember that geometric shapes have well-established properties and definitions, and that shapes belong to specific geometric classes, which in themselves have specific postulates, properties, definitions and theorems associated with them.

5. Study existing two-column proofs in textbooks and examine the reasons in the reason columns and the statements in the statement column in each proof. Write each type of reason you find in a reason book. Note whether the reason is a postulate, definition, property or theorem. Note any reasons you are not familiar with and research these reasons so as to validate them as reasons. In each two-column proof, consider each statement associated with each reason. Relate the statement to the reason and understand why the reason makes the statement true. Use and refer to your reason book and existing proofs to help you establish a plan to prove your proof.

References

• SparkNotes: Geometric Proofs: Terms
• Oswego City School District Regents Exam Prep Center: Styles of Proof
• HomeSchoolMath: Example of two-column proof vs. paragraph proof
• University of New Mexico: Deductive Proof in Geometry by William Glover
• OnMathLearning: Geometry Proofs--CPCTC, Two-Column Proofs, FlowChart Proofs