1. 1.7 DEDUCTIVE STRUCTURE
Geometry is based on deductive structure. Using deductive structure, we justify
conclusions using previously assumed or proved statements.
The deductive structure uses:
 undefined terms - These are words that we use that do not have a precise
definition, but people have agreed as to what they mean. Some
examples of such terms are point and line. We have never given a precise
definition of a point, but everyone seems to accept what a point is.
 definitions – Definitions state the meaning of terms or ideas.
 assumptions (postulates) – These are unproven assumptions – statements
that we accept as truth and use in proofs.
 theorems (or conclusions) – These are mathematical statements that can
be proved. We have used theorems several times already.
Reversibility:
Definitions are reversible. In order to understand what that means, look at the
following example.
 If a point is the midpoint of a segment, then the point divides the segment
into 2 congruent segments.
 If a point divides a segment into 2 congruent segments, then the point is
the midpoint of the segment.
Both definitions of a midpoint are correct; however, one may fit better than the
other in a proof.
Notice these definitions are in the form “If this, then that.” or “If p, then q.” This is
called a conditional statement.
The “if” part is called the hypothesis.
The “then” part is called the conclusion.
There is more than one way to write “If p, then q.” You can say “p implies q” or
“p⇒q.”

2. When you write the statement in reverse order (“If q, then p.”), it is called the
converse.
Though definitions are reversible, theorems and postulates are not always
reversible.
For example, consider the theorem: “If 2 angles are right angles, then they are
congruent.”
What is the converse of this?
“If 2 angles are congruent, then they are right angles.”
The converse is a crazy, wild, false assumption!