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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.