The most common form of proof in geometry is direct proof. In a direct
proof, the conclusion to be proved is shown to be true directly as a result of
the other circumstances of the situation. The sample proof from the previous
lesson was an example of direct proof. In that previous, the triangles were
shown to be congruent directly as a result of their sharing two equal
corresponding sides and one equal included angle.
Direct proof is deductive
reasoning at
work. Throughout a direct proof, the
statements that are made are specific
examples of more general situations, as is explained in the "reasons" column.
For example, you might state in a direct proof that two angles sum to 90
degrees, and in the next line, state that they are complementary. In the
reasons, column, you would write "Two angles whose sum is 90 degrees are
complementary." Think of this reason as a conditional
statement "if p, then q" where p is "angles sum to 90 degrees" and q is "they are complementary." The first statement, that the angles sum to 90 degrees, is the
hypothesis of the conditional statement. If
the conditional statement is true, which we know it is, then q, the next
statement in the proof, must also be true. In this example, the conditional
statement is a general statement about angles, and using it, it is deduced that
a specific pair of angles is complementary. In this way, direct proof makes use
of deductive reasoning.
For now, direct proof is the only kind of proof we know, but in a few lessons,
we'll learn about indirect proof, in which the conclusion to be proved is
shown to be true because every other possibility leads to a contradiction.