Non-Euclidean Geometry

Br. Joel Baumeyer, FSC

Christian Brothers University

The Axiomatic Method of Proof 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 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

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 and are between A and B. (A and B are endpoints.)

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Euclid’s First Four Axioms (2) and necessary definitions

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

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 is the following set of points lying on the line : those points that belong to the segment AB and all points C on such that B is between A and C. The ray is said to emanate from the vertex A and to be a part of line .

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Euclid’s First Four Axioms (4) and necessary definitions

Def: Rays and are opposite if they are distinct, if they emanate from the same point A, and if they are part of the same line = .

Def: An "angle with vertex A" is a point A together with two distinct non-opposite rays and , called the sides of the angle emanating from A.

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Euclid’s First Four Axioms (5) and necessary definitions

Def: If two angles BAD and CAD have a common side and the other two sides and 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.

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The Parallel Postulate

Def: Two lines l and m are parallel if they do not intersect, i.e. if no point lies on both of them.

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 AxiomsA-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 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.