Euclid's Geometry: Undefined Terms & Postulates

• Chapter 2 discusses undefined terms in Euclid.

Euclid's Postulates

• A straight line can be drawn from any point to any point.
• A straight line can be produced continuously in a straight line.
• A circle can be described with any center and radius.
• All right angles are equal to one another.
• If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.
• Euclid's plane geometry presentation was flawed in at least three ways because it failed to recognize undefined terms, used unstated postulates in theorem proofs, and relied on diagrams for proof logic.
• Only one line can be drawn parallel to a given line through a given point.
• There is at least one triangle where the sum of the interior angles is 180°.
• Parallel lines are always equidistant.
• Two triangles exist that are similar but not congruent.
• Two straight lines exist that are equidistant at three points.
• Every triangle can be circumscribed.
• The sum of the measures of the interior angles of a triangle is the same for all triangles.
• Rectangles can be constructed using a compass and a straightedge.
• Playfair's postulate.

Hilbert's Axioms Overview

• These are for Euclidean Plane Geometry and cover undefined terms.

Undefined Terms

• Point
• Line
• Plane
• Lie (incidence of point and line)
• Between
• Congruence

Axioms of Incidence (Connection):

• For every two points A and B, there exists a line that contains each of the points A and B.
• For every two points A and B, there is only one line that contains both points A and B.
• There exist at least two points on a line; there exist at least three points that do not lie on a line.
• For any three non-collinear points A, B, C, there exists a plane α that contains A, B, C; every plane contains at least one point.

Axioms of Order include:

• If point B is between points A and C, then A, B, and C are distinct points on the same line, and B is between C and A.
• For any two distinct points A and C, there is at least one point B on line AC where C is between A and B.
• If A, B, and C are three points on the same line, no more than one is between the other two.
• Given three non-collinear points A, B, C, if a line l in the plane of A, B, C doesn't pass through A, B, or C, and it intersects segment AB, then it must also intersect segment AC or segment BC.

Axioms of Congruence

• If A and B are two points on a line a, and A' is a point on line a', a point B' can be found on a given side of a' where segment AB is congruent to A'B'.
• Segments A'B' and A"B" congruent to the same segment AB are congruent to each other.
• On a line a, segments AB and BC share no common points besides B; on another line a', segments A'B' and B'C' share no common points besides B'.
• If AB = A'B' and BC = B'C', then AC = A'C'.
• Given angle ∠ABC and ray B'C', there exists exactly one ray B'A' on each side of B'C' such that ∠A'B'C' = ∠ABC.
• Every angle is congruent to itself.
• If ∆ABC and ∆A'B'C' have AB = A'B', AC = A'C', and ∠BAC = ∠B'A'C', then ∠ABC = ∠A'B'C'.

Axiom of Parallels

• Given a line a and a point A not on it, there is at most one line in the plane containing a and A that passes through A and does not intersect a.

Axioms of Continuity

• Archimedes Axiom: For any segments AB and CD, there exists a number n such that n copies of CD, constructed contiguously from A along ray AB, will pass beyond point B.
• Axiom of Line Completeness: It is impossible to extend a set of points on a line with its order and congruence relations while preserving the existing relations and fundamental properties from Axioms I-III and V-1.

Birkhoff's Postulates for Euclidean Plane Geometry

• Defined in terms of undefined terms and relations.

Undefined Terms and Relations

• Points
• Sets of points called lines
• Distance between any two points A and B: a nonnegative real number d(A, B), such that d(A, B) = d(B, A)
• Angle formed by three ordered points A, O, B (A ≠ 0, B ≠ 0; ∠AOB such that m∠AOB is a real number (mod 2π)

Definitions

• Point B is between A and C (A ≠ C) if d(A, B)+d(B, C) = d(A, C).
• The points A and C with all points B between them form line segment AC.
• The half-line m' with endpoint O is defined by two points O, A in line m(A ≠ 0) as the set of all points A' of m where O is not between A and A'.
• Two distinct lines with no common points are parallel, and a line is considered parallel to itself.
• Half-lines m and n through O form a straight angle if mLmOn = (+/-)π, and a right angle if mLmOn = (+/-)π/2, in which case m is perpendicular to n.
• Three distinct points A, B, C form a triangle ∆ABC with sides AB, BC, CA and vertices A, B, C; if A, B, and C are collinear, ∆ABC is degenerate.
• Any two geometric figures are similar if there exists a one-to-one correspondence between the points of the two figures such that all corresponding distances are in proportion and corresponding angles have equal measures (except, perhaps, for their sign). Any two geometric figures are congruent if they are similar with a constant of proportionality k = 1.

Postulates include:

• Postulate of Linear Measure: The points A, B, ..., of any line m can be placed into a one-to-one correspondence with the real numbers r so that |rB - rA| = d(A, B) for all points A and B.
• Point-line Postulate: One and only one line m contains two given points P and Q (P ≠ Q).
• Postulate of Angular Measure: The half-lines m, n, ..., through any point O can be placed into a one-to-one correspondence with real numbers α (mod 2π) so that if A ≠ 0 and B≠ 0 are points of m and n, respectively, the difference (αn - αm) (mod 2π) is m∠AOB.
• Postulate of Similarity: If ∆ABC and A'B'C', and for some positive constant, k, d(A', B') = kd(A, B), d(A', C') = kd(A, C), and also m∠BAC = (±)m∠B'A'C', then also d(B', C') = kd(B, C) and m∠C'B'A' = (±)<CBA and m∠A'C'B' = (±)m∠ABC.

The SMSG Postulates for Euclidean Geometry

Undefined Terms

• Point
• Line
• Plane

Postulates include:

• Given any two distinct points, there is exactly one line that contains them.
• Distance Postulate: Every pair of distinct points corresponds to a unique positive number, termed the distance between the two points.
• Ruler Postulate: The points of a line can correspond with real numbers such that every point corresponds to one real number, every real number corresponds to one point on the line, and the distance between points is the absolute value of the difference of their real numbers.
• Ruler Placement Postulate: For two points P and Q on a line, the coordinate system can be set so P is zero and Q is positive.
• Every plane contains at least three noncollinear points, and space contains at least four noncoplanar points.
• If two points lie in a plane, the line containing them also lies in that plane.
• Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane.
• If two planes intersect, their intersection is a line.
• Plane Separation Postulate: Given a line and a plane containing it, the points of the plane not on the line form two convex sets such that any segment connecting a point from each set intersects the line.
• Space Separation Postulate: The points of space not lying in a given plane form two convex sets such that any segment connecting a point from each set intersects the plane.
• Angle Measurement Postulate: Each angle corresponds to a real number between 0° and 180°.
• Angle Construction Postulate: Given ray AB on the edge of half-plane H, for every r between 0° and 180°, there is exactly one ray AP with P in H such that m∠PAB = r.
• Angle Addition Postulate: If D is in the interior of ∠BAC, then m∠BAC = m∠BAD + m∠DAC.
• Supplement Postulate: If two angles form a linear pair, they are supplementary.
• SAS Postulate: If two triangles (or a triangle with itself) have two sides and the included angle of the first congruent to the corresponding parts of the second, the correspondence is a congruence.
• Parallel Postulate: At most one line can be drawn through an external point parallel to a given line.
• Every polygonal region corresponds to a unique positive real number called its area.
• If two triangles are congruent, their triangular regions have the same area.
• If region R is the union of regions R1 and R2, which intersect at most in a finite number of segments and points, then the area of R equals the sum of R1 + R2.
• The area of a rectangle is the product of its base length and altitude length.
• The volume of a rectangular parallelepiped is the product of its altitude length and base area.
• Cavalieri's Principle: If two solids are cut be every plane parallel to a given plane, and these intersections have the same area, then both solids have the same volume.

Non-Euclidean Geometries

Hyperbolic Parallel Postulate

• Given a line l and point P, there are at least two distinct lines through P parallel to l.
• An infinite number of parallels to a given line can be drawn through a given point.
• No triangles exist where the sum of the angle measures is 180°.
• No similar but not congruent triangles exist.
• No lines are everywhere equidistant.
• There exist triangles that cannot be circumscribed.
• The sum of interior angles of triangles varies and is always less than 180°.
• Rectangles do not exist.
• There is an upper limit to a triangle's area.
• Larger triangles have smaller angle sums.
• The distance between certain parallel lines approaches zero in one direction and infinity in the other.
• Alternate interior angles may not be congruent if two parallel lines are crossed by a transversal.

Elliptic Parallel Postulate

• Given a line l and a point P not on l, there is no line through P parallel to l.
• Hyperbolic geometry: angle sum < 180°.
• Euclidean geometry: angle sum = 180°.
• Elliptic geometry: angle sum > 180°.