Axioms of Inequality
Just as axioms exist for equality, similar axioms exist for inequality. The only axiom of equality that
has no counterpart for inequality is the reflexive axiom. The other seven are as follows.
The Transitive Axiom
PARGRAPH The transi
Deductive Reasoning
Deductive reasoning, unlike inductive reasoning, is a valid form of proof. It is, in fact, the way in
which geometric proofs are written. Deductive reasoning is the process by which a person makes
conclusions based on previously known
Axioms of Equality
In this section, we will outline eight of the most basic axioms of equality.
The Reflexive Axiom
The first axiom is called the reflexive axiom or the reflexive property. It states that any quantity is
equal to itself. This axiom governs
Building Blocks of Proof
Just as figures in a plane are made of building blocks such as points, segments, and lines,
geometric proofs are made of building blocks, too. These building blocks include definitions,
postulates, axioms, and theorems. Together,
Auxiliary Lines
Often in a proof, it becomes helpful to modify the figure that you are given. It is never acceptable to
change any of the original parts of the figure, but it is appropriate to draw in new lines that will help
demonstrate something. Such l
Postulates
A number of postulates have to do with lines. Some are listed here.
Through any two points, exactly one line can be drawn.
Two lines can intersect at either zero or one point, but no more than one.
Through a point not on a line, exactly one
Variations on Conditional Statements
The three most common ways to change a conditional statement are by taking its inverse, its
converse, or it contrapositive. In each case, either the hypothesis and the conclusion switch places,
or a statement is replac
Inductive Reasoning
Inductive reasoning is the process of arriving at a conclusion based on a set of observations. In
itself, it is not a valid method of proof. Just because a person observes a number of situations in
which a pattern exists doesn't mean t
Variations Using Statements
Negations
Every statement has a negation. Usually the negation of a statement is simply the same statement
with the word "not" before the verb. The negation of the statement "The ball rolls" is "The ball does
not roll." By defi
Statements
Declarative Sentences
As the Introduction said, geometry consists of numerous declarative sentences. A declarative
sentence is a sentence that asserts the truth or falsehood of something. For example, "That car is
red" is a declarative sentence
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