TOC & Ch. 0 & Ch. 1 Axiom Table of Contents Ch. 0 Introduction Ch. 1 Axiomatic Systems 1.1.1 Introduction 1.1.2 Examples 1.1.3 History 1.2 A Finite Geometry 1.3 Finite Projective 1.4 Applications

Ch. 2 Neutral Geometry Ch. 2 Table of Contents 2.1.1 Introduction 2.1.2 History 2.1.3 Analytic Models 2.2 Incidence Axioms 2.3 Distance/Ruler Axioms 2.4.1 Plane Separation Axiom 2.4.2 Angle & Measure 2.5.1 Supplement Postulate 2.5.2 SAS Postulate 2.6.1 Parallel Lines 2.6.2 Saccheri Quadrilateral 2.7.1 Euclid Parallel Postulate 2.7.2 Hyperbolic Parallel Postulate 2.7.3 Elliptic Parallel Postulate 2.8 Euclid/Hyperbolic/Elliptic Birkhoff's Axioms Hilbert's Axioms SMSG Axioms

Ch. 3 Transformational Ch. 3 Table of Contents 3.1.1 Introduction 3.1.2 History 3.2.1 Definitions 3.2.2 Analytic Model 3.2.3 Affine Transformation 3.3.1 Isometry 3.3.2 Model/Collinearity 3.3.3 Model/Isometry 3.4.1 Direct Isometry 3.4.2 Model/Direct 3.5.1 Indirect Isometry 3.5.2 Model/Indirect 3.6.1 Similarity Transformation 3.6.2 Model/Similarity 3.7 Other Affine Transformations

Ch. 4 Projective Geometry Ch. 4 Table of Contents 4.1.1 Introduction 4.1.2 Historical 4.2.1 Axioms 4.2.2 Basic Theorems 4.3 Duality 4.4 Desargue's Theorem 4.5.1 Harmonic Sets 4.5.2 Music & Harmonic Sets 4.6.1 Definitions for Projectivity 4.6.2 Fundamental Theorem 4.6.3 Projectivity/Harmonic Sets 4.6.4 Alternate Construction 4.7.1 Conics 4.7.2 Pascal's Theorem 4.7.3 Tangents to Conics

Other Topics Ch. 5 Spherical Geometry Ch. 6 Fractal Geometry Ch. 7 Topology

Appendices Internet Resources Index Geometer's Sketchpad/GeoGebra JavaSketchpad/GeoGebraHTML Video Lectures Logic Review References Acknowledgements

          2.5.2 SAS Postulate  Printout
Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions.
 —Eric Temple Bell (1883–1960)

 

Postulate 15. (SAS Postulate) Given a one-to-one correspondence between two triangles (or between a triangle and itself). If two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence.

      We restate the Crossbar Theorem here since it plays an important role in the proofs of some of the results in this section.

Theorem 2.9. (Crossbar Theorem) If , then ray BP and segment AC intersect in a unique point F and A-F-C.

Definition.
      An isosceles triangle is a triangle with two congruent sides. If the isosceles triangle has exactly two congruent sides, the angles opposite the two congruent sides are called base angles, the angle formed by the two congruent sides is called the vertex angle, and the third noncongruent side is called the base.
      An equilateral triangle is a triangle with all sides congruent.
      A scalene triangle has no congruent sides.

Theorem 2.10. (Pons Asinorum) The base angles of an isosceles triangle are congruent.

      Pons Asinorum (Bridge of Asses) is Proposition 5 from Book 1 of Euclid's Elements. The name comes from the diagram, which looks like a bridge, used in Euclid's method for proving the theorem. (See the figure on the right or Byrne's Edition of Euclid's Elements.) The method used here is similar to the method used in many high school courses with one major exception. Since a proof should not be based on a picture and preconceived ideas, we need the Crossbar Theorem to fill in a gap that is not addressed in most high school courses.

Proof. Let  be an isosceles triangle with . Since every angle has a unique angle bisector, let ray AD be the bisector of . By the Crossbar Theorem, ray AD and segment BC intersect at a unique point E and B-E-C. Thus . Since congruence of segments is an equivalence relation, . Hence by SAS Postulate, . Thus . Therefore, the base angles of an isosceles triangle are congruent.//

Exercise 2.48. Find the axiom from a high school book that corresponds to the SAS Postulate.

Exercise 2.49. Show the SAS Postulate is not satisfied by the (a) Taxicab plane; and (b) Max-distance plane. Thus showing independence.

Exercise 2.50. Prove or disprove. If quadrilateral ABCD is such that BD and AC intersect at a point M and M is the midpoint of both BD and AC, then AB is congruent to CD.

Exercise 2.51. Prove or disprove. If quadrilateral ABCD has CD congruent to CB and ray CA is the bisector of , then AB is congruent to AD.

Exercise 2.52. Prove or disprove. If  is a quadrilateral and , then .

Exercise 2.53. State and prove each theorem.  (a) SSS Theorem; and  (b) ASA Theorem.

Exercise 2.54. Prove that all points equidistant from two points A and B are on the perpendicular bisector of AB.