Geometry Honors · Lesson 7.3
Axiomatic Systems
How it works
An axiomatic system begins with undefined terms, definitions, and accepted postulates. Theorems follow by deduction. A proof must use the chosen system: Euclidean parallel-line results need not hold on a sphere.
Worked examples
Example 1
Distinguish a definition from a theorem using midpoint.
A midpoint is defined to divide a segment into two congruent segments. A claim proved using that definition is a theorem or consequence.
Example 2
Why is a diagram insufficient to establish a theorem?
A theorem applies to every figure satisfying the hypotheses; one drawing does not establish all cases.
Problems
Problem 7.3.1
Identify the assumption in: Through two distinct points there is exactly one line.
Show worked solution
This is an incidence postulate in Euclidean geometry, accepted as a starting fact rather than proved from a picture.
Problem 7.3.2
Can the sum of triangle angles be assumed to be in every geometry?
Show worked solution
No. The Euclidean result depends on the geometric system. Spherical triangles can have larger angle sums.
Problem 7.3.3
Why is using a conclusion as a premise invalid in a proof?
Show worked solution
It is circular: it assumes the very fact that the argument is supposed to establish.