5. Introduction to Euclid’s Geometry
- 1. NUMBER SYSTEMS
- 2. A POLYNOMIALS
- 3. COORDINATE GEOMETRY
- 4. LINEAR EQUATIONS IN TWO VARIABLES
- 5. INTRODUCTION TO EUCLID'S GEOMETRY
- 6. LINES AND ANGLES
- 7. TRIANGLES
- 8. QUADRILATERALS
- 9. CIRCLES
- 10. HERON'S FORMULA
- 11. SURFACE AREAS AND VOLUMES
- 12. STATISTICS
- 13. APPENDIX 1: PROOFS IN MATHEMATICS
- 14. APPENDIX 2: INTRODUCTION TO MATHEMATICAL MODELLING
5.1 Introduction - Learning Objectives
5.2 Euclid’s Definitions, Axioms and Postulates
Example 1:
If A, B and C are three points on a line and B lies between A and C, prove that AB + BC = AC.
Solution: Use Euclid’s Axiom (4): Things which coincide with one another are equal to one another.
Example 2:
Prove that an equilateral triangle can be constructed on any given line segment.
Solution: Use Euclid’s Postulate 3 (draw circles) and Axiom (1) (equality of radii).
Theorem 5.1
Two distinct lines cannot have more than one point in common.
Proof: Assuming two lines intersect at more than one point leads to contradiction with Axiom 5.1 (unique line through two points).
Exercise 5.1
- Determine truth of basic geometric statements and justify answers.
- Define key geometric terms and identify which terms must be defined first.
- Discuss consistency and relation of given postulates to Euclid's postulates.
- Prove: If C lies between A and B, and AC = BC, then AC = ½AB.
- Prove: Every line segment has one and only one midpoint.
- Prove: If AC = BD, then AB = CD in a geometric figure.
- Explain why Axiom 5 is considered a universal truth.