Modern Geometry

  1. Read the attached section from the Venema book regarding different axiomatic systems – Hilbert’s Axioms,
    Birkhoff’s Axioms, Maclane’s Axioms, and SMSG axioms.
  2. Compare each of these 4 axiomatic systems with the Neutral Geometry system that we had this semester.
    In particular, you should specify how each of their axioms relate to our axioms and theorems. Whenever
    possible, prove that an axiom can be obtained from our axiomatic system. You can ignore axioms that
    relate to areas and volumes.
  3. Discuss the advantages and disadvantages of each axiomatic system. Give specific examples.
    This project should be about 5 pages. Typed is preferred, but a paper written by hand clearly and legibly is also
    acceptable, as long as you submit clear and legible photos of it.
  4. Axiom 1: The Existence Postulate (a1) The collection of all points forms a non-empty set. There is more than one point in that set. Axiom 2: The Incidence Postulate (a2) Every line is a set of points. For every pair of distinct points A and B there is exactly one line l = !AB such that A 2 l and B 2 l. Axiom 3: The Ruler Postulate (a3) For every pair of points P and Q there exists a real number P Q, called the distance from P to Q. For each line l there is a one-to-one correspondence from l to R such that if P and Q are points on the line that correspond to the real numbers x and y, respectively, then P Q = jx yj. Axiom 4: The Plane Separation Postulate (a4) For every line l, the points that do not lie on l form two disjoint, non-empty sets H1 and H2, called half-planes bounded by l, such that the following conditions are satisÖed: 1. Both H1 and H2 are convex 2. If P 2 H1 and Q 2 H2, then P Q intersects l Axiom 5: The Protractor Postulate (a5) For every angle \BAC there is a real number  (\BAC), called the angle measure of \BAC; such that the following conditions are satisÖed 1. 0    (\BAC) < 180 for every angle \BAC 2.  (\BAC) = 0 if and only if !AB = !AC 3. For each real number r, 0 < r < 180; and for each half-plane H bounded by !AB there exists a unique ray !AE such that E is in H and  (\BAE) = r  4. If the ray !AD is between rays !AB and !AC then  (\BAD) +  (\DAC) =  (\BAC) Axiom 6: The SAS Postulate If ABC and DEF are two triangles such that AB = DE; BC = EF and \ABC = \DEF, then ABC = DEF: DeÖnition 1 Two lines l and m are said to intersect if l \ m 6= ?: DeÖnition 2 Two lines l and m are said to be parallel if l \ m = ?: DeÖnition 3 Three points A; B; C are said to be collinear if there exists one line l such that A; B; C all lie on l. DeÖnition 4 Let A; B; C be distinct points. The point C is said to be between A and B, written ACB, if A; B; C are collinear and AC + C DeÖnition 5 DeÖne segment AB by AB = fA; Bg [ fPjA P Bg and the ray !AB by !AB = AB [ fPjA B Pg The points A and B are said to be the endpoints of AB and A is the endpoint of the ray !AB. DeÖnition 6 The length of a segment AB is the distance AB between the endpoints A and B. DeÖnition 7 Two segments AB and CD are said to be congruent, written AB = CD; if they have the same length. DeÖnition 8 A set of points S is said to be a convex set if for every pair of points A and B in S, the entire segment AB is contained in S. DeÖnition 9 Two angles \BAC and \DEF are called congruent if  (\BAC) =  (\DEF): DeÖnition 10 Two angles \BAC and \DEF are called supplements if  (\BAC) +  (\DEF) = 180 : DeÖnition 11 If l; l0 are two distinct lines, a line t is said to be a transversal thats cuts l and l 0 if t intersects both l and l 0 : DeÖnition 12 Suppose l and l 0 are two lines cut by a transversal t. Suppose t intersects l at the point B and l 0 at the point B0 : Suppose A and C are two points on l such that A B C and A0 ; C0 are two points on l 0 such that A0 B0 C 0 and A0 is on the same side of t as A and C is on the same side of t as C: Then, the angles \A0B0B and \B0BC are called alternate interior angles. DeÖnition 13 In the same setup as in the above deÖnition, suppose B00 is a point on t such that B B0 B00 : Then, the angles \B00B0C 0 and \B0BC are called corresponding angles. DeÖnition 14 Suppose A; B; C; D are four distinct points, no three of which are collinear. Suppose that any two of the segments AB; BC; CD; and DA either have no points in common or have only an endpoint in common. Then, the vertices A; B; C; D determine a quadrilateral ABCD = AB [ BC [ CD [ DA: DeÖnition 15 A quadrilateral ABCD is said to be a convex quadrilateral if each vertex is contained in the interior of the angle formed by the other three vertices. DeÖnition 16 A Saccheri quadrilateral is a quadrilateral ABCD such that \ABC and \DAB are right angles and AD = BC: The segment AB is called the base and the segment CD is called the summit. The two right angles \ABC and \DAB are called base angles and the two angles \CDA and \BCD are called summit angles. DeÖnition 17 A parallelogram is a quadrilateral ABCD such that !AB k !CD and !AD k !BC: Theorem 1 (Existence and uniqueness of midpoints) Let AB be a segment, such that A and B are distinct. Then, there exists a unique point M 2 AB such that AM = MB, called the midpoint. Theorem 2 (The Ray Theorem) Let l be a line, A a point of l and B an external point for /l. If C is a point on !AB and C 6= A, then B and C are on the same side Theorem 3 Let A; B and C be three noncollinear points and let D be a point on the line !BC: The point D is between points B and C if and only if the ray !AD is between rays !AB and !AC: Theorem 4 (Betweenness Theorem for Rays) Let A; B; C; D be four distinct points such that C and D lie on the same side of !AB: Then  (\BAD) <  (\BAC) if and only if !AD is between rays !AB and !AC Theorem 5 (Linear Pair Theorem) If angles \BAD and \DAC form a linear pair, then  (\BAD)+  (\DAC) = 180 : Theorem 6 (Vertical Angles Theorem) Vertical angles are congruent. Theorem 7 (Exterior Angle Theorem) The measure of an exterior angle for a triangle is strictly greater than the measure of either of the remote interior angles. Theorem 8 If ABC is a triangle, DE is a segment such that DE = AB and H is a half-plane bounded by !DE, then there is a unique point F 2 H such that DEF = ABC: Theorem 9 (Isosceles Triangle Theorem) The base angles of an isosceles triangle are congruent. Theorem 10 In any triangle, the greater side lies opposite the greater angle and the greater angle lies opposite the greater side. Theorem 11 (Alternate Interior Angles) If l and l 0 are two lines cut by a transversal t such that the alternate interior angles are congruent, then l k l 0 : Theorem 12 If ABCD is a convex quadrilateral, then  (ABCD  360 ): Theorem 13 Every parallelogram is a convex