Transcript Ch_1_ 10f - Christian Brothers University

Non-Euclidean Geometry
Br. Joel Baumeyer, FSC
Christian Brothers University
The Axiomatic Method of Proof
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The axiomatic method is a method of
proving that a conclusion is correct.
Requirements:
0. The meaning of words and symbols used
in the discussion must be understood
clearly by all involved in the discussion.
1. Certain statements called axioms (or
postulates) are accepted without
justification.
2. Agreement on certain rules of reasoning
(I.e. agreeing on how and when one
statement follows another).
Euclid’s Axiom Base
Terms
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Defined terms - words which have
common meaning to all involved in the
discussion and which are subject to the
rules of reasoning in requirement # 2
above.
Undefined terms - words which are not
defined but to which certain properties
are ascribed. For Euclidean plane
geometry they are:
– point, line, (plane)
– lie on, between, congruent
Some Basic Concepts
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A Set is a collection of objects.
Equal vs Congruent*
– Equal (=) means identical ()
– Congruent( ) undefined (for the time being think
“fits exactly on”)
*both terms are what is known as equivalence
relations
Euclid’s First Four Axioms
(1)
and necessary definitions
P1: For every point P and for every point
Q not equal to P there exists a unique
line l that passes through P and Q.
Def: The segment AB is the set whose
members are the points A and B and all
points that line on the line AB and are
between A and B. (A and B are
endpoints.)
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Euclid’s First Four Axioms
(2)
and necessary definitions
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P2: For every segment AB and for every
segment CD there exists a unique point
E such that B is between A and E and
segment CD is congruent to segment
BE.
Def: Given two points O and A. The set
of all points P such that segment OP is
congruent to segment OA is called a
circle with O as center and each
segment OP is called a radius of the
circle.
Euclid’s First Four Axioms
(3)
and necessary definitions
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P3: For every point O and every point A
not equal to O there exists a circle with
center O and radius OA
Def: The ray AB is the following set of
points lying on the line AB : those
points that belong to the segment AB
and all points C on AB such that B is
between A and C. The ray AB is said
to emanate from the vertex A and to be
a part of line AB .
Euclid’s First Four Axioms
(4)
and necessary definitions
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Def: Rays AB and AC are opposite if
they are distinct, if they emanate from
the same point A, and if they are part
of the same line AB = AC .
Def: An "angle with vertex A" is a point
A together with two distinct nonopposite rays AB and AC, called the
sides of the angle emanating from A.
Euclid’s First Four Axioms
(5)
and necessary definitions
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Def: If two angles BAD and CAD
have a common side AD and the other
two sides AB and AC form opposite
rays, the angles are called
supplementary angles.
Def: An angle
BAD is a right angle if
it has a supplementary angle to which it
is congruent.
P4: All right angles are congruent to
each other.
The Parallel Postulate
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Def:
Two lines l and m are parallel
if they do not intersect, i.e. if no point
lies on both of them.
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Euclid’s Parallel Axiom: For every
line l and for every point P that does
not lie on l there exists a unique line
m through P that is parallel to l.
Euclid’s First Five Axioms
A-1: For every point P and for every point Q not equal to P there
exists a unique line l that passes through P and Q.
A-2: For every segment AB and for every segment CD there exists
a unique point E such that AB is between A and E and segment CD
is congruent to segment BE.
A-3: For every point O and every point A not equal to O there
exists a circle with center O and radius OA.
A-4: All right angles are congruent to each other.
A-5: For every line l and for every point P that does not lie on l
there exists a unique line m through P that is parallel to l.
Euclid’s First Five Axioms
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A-1: For every point P and for every point Q not
equal to P there exists a unique line l that passes
through P and Q.
A-2: For every segment AB and for every segment
CD there exists a unique point E such that AB is
between A and E and segment CD is congruent to
segment BE.
A-3: For every point O and every point A not equal to
O there exists a circle with center O and radius OA.
A-4: All right angles are congruent to each other.
A-5: For every line l and for every point P that does
not lie on l there exists a unique line m through P that
is parallel to l.
Legendre’s Parallel Proof: