This section introduces the student to the subject of Geometry through the study of undefined terms, defined terms and the resulting axioms. It explains the axiomatic approach to studying geometry and how theorems can be derived from axioms.

Section 1: Objectives

Students will:

1. be able to explain how geometry is related to nature.
2. be able to describe undefined and defined terms.
3. be able to explain how axioms can be stated based on these terms.
4. be able to explain how theorems are derived with the help of axioms.

Topic 1.1  Structure of Geometry – Axiomatic Approach

Introduction to Geometry:

The word “Geometry” is derived from the Greek language. The word “Geometry” is divided into two “Geo” and “Metry”. “Geo” means “Earth”, “Metry” means “Measure”.

Therefore, Geometry means measurement of magnitude of space or planes like area, volume, etc.

The credit of building up Geometry goes to Euclid, the great Greek Mathematician of 3rd Century B.C.

All sciences have evolved from nature and so is mathematics and geometry in particular. As such the structure of nature should be understood first before we try to understand the structure of geometry.

Nature consists of a number of animate and inanimate objects. These are usually trees, flowers, fruits, rivers, earth, moon, sun, etc. These are basic objects of nature and do not require definition. We call them as undefined objects. With the help of these undefined objects, terms are created. For example, with the help of an undefined object, say Sun, we define the term east as the direction of the sun rises every day and west as the direction where the Sun sets every day.

Having chosen such undefined and defined terms, certain factual statements have been formed. For example, taking the undefined term Sun and defined terms east and west, statements like “The Sun rises in the east and sets in the west” have been made. This statement is self-evident fact, which is true and does not require any proof or explanation. Such statements are called Axioms.

Let us study geometry by studying the axioms.

The structure of geometry spotlights the following essential stages namely:

     1. Undefined and defined terms.
     2. Axioms.
     3. Theorems.
     4. Induction and deductions.

These essential stages will be learned in the next topics.