Axiomatic Development of Geometry: An Introduction

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AXIOMATIC DEVELOPMENT
OF GEOMETRY
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13 14 15 16 It is a system made up
undefined terms, defined
terms, axioms/postulates, and
theorems which are used in
proving logical conclusions in
geometry.
Axiomatic
system
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3. Undefined terms
Defined terms
axioms/postulates
theorems
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13 14 15 16 Are terms that cannot be
defined because they can only
be described or illustrated.
Undefined
terms
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The three undefined terms are
point, line and plane.

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Are terms with a precise and
concise definition.
defined terms
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A definition is a precise
statement or description of the
meaning of the term or word so
that anyone using it will
understand it in the same way.
Ex. Parallel lines, angle,
midpoint

6. defined terms
■ Angle - a figure formed by two rays called the sides of the angle, sharing
a common endpoint, called the vertex of the angle.
■ Parallel Lines - lines in a plane which do not meet.
■ Collinear Points - points that lie on the same line.
■ Coplanar Points - points that lie on the same plane.

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Are statements accepted to be
true without proof.
Postulates are statement from
geometry.
Axioms are statement from
other sections of mathematics.
axioms/postulates
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8. postulate
■ Line postulate - for every two points, there is exactly one line that
contains both points.
■ Plane postulate - any three noncollinear points lie in at least one plane.

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Are statements that are
proven to be true using
definitions, axioms/postulates,
and derived using reasoning.
theorem
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Proof - a sequence of true facts
that are arranged in a logical
order.

10. theorem
■ Vertical Angles Theorem: Vertical angles are equal in measure.

11. theorem
■ Line-Point-Plane Theorem - given a line and a point, there is exactly one
plane containing both the line and the point.

12. theorem
■ Proof:
■ Points M, O and C are non collinear points. By plane postulate, any
three noncollinear points lie exactly in one plane. Also, by line
postulate, any two points determine a straight line. Therefore, there is
exactly one plane containing line MO and point C.