Geometry: Logic Statements
Applying Logic Statements to Geometry
As we study statements like "If the sun shines, then the grass will grow," it is easy to lose focus of geometry and the purpose of studying logic statements at all. The reason to become familiar with logic statements is to understand the definitions of geometric figures and terms so that they may be properly used in geometric proofs. Geometric proofs are displays of irrefutable lines of reasoning by which we can show certain things to be true beyond doubt. If a definition is improperly used or too much is assumed of a given figure, the proof is worthless.
Perhaps, in a problem, you will be given a quadrilateral and told that the opposite angles are congruent. You think that the quadrilateral might be a parallelogram, but can you be sure? The questions you ask yourself are 1) Are the opposite angles of a parallelogram always congruent?, and 2) Are there any other figures whose opposite angles are congruent? What you are actually doing is checking the truth of a statement and its converse. The first question you ask yourself translates to this statement: If a quadrilateral is a parallelogram, then its opposite angles are congruent. The second question translates to the converse of the previous statement: If the opposite angles of a quadrilateral are congruent, then it is a parallelogram. Hopefully in this situation you would realize that both the statement and its converse are true, meaning that either statement is a valid definition for parallelograms, and the figure in question definitely is a parallelogram.
Relationships like this exist all throughout geometry. It is not our ultimate goal to be able to draw a perfect truth table with 1,000 columns and one million rows! All we need to know is how to properly use and test definitions, so that we don't mislabel a figure in a proof. In some proofs, all that you'll be given is a drawing, and from it, you must figure out what kind of geometric figure it is. Remember: the process of deductive reasoning is only good if every step of the process is done correctly. When this happens, the conclusion is irrefutable, but when even one conclusion drawn is not entirely valid (i.e. a parallelogram was assumed to be a rhombus), then the entire line of reasoning is faulty and in the end, worthless. Hopefully with an understanding of logic statements, every step you take will be a step in the right direction.