Chapter 5. INTRODUCTION TO EUCLID’S GEOMETRY

1. The word ‘geometry’ comes form the Greek words ‘geo’, meaning the ‘earth’,

and ‘metrein’, meaning ‘to measure’. Geometry appears to have originated from

the need for measuring land.

2. Theorem 5.1 : Two distinct lines cannot have more than one point in common.

Proof : Here we are given two lines l and m. We need to prove that they have only one

point in common.

For the time being, let us suppose that the two lines intersect in two distinct points,

say P and Q. So, you have two lines passing through two distinct points P and Q. But

this assumption clashes with the axiom that only one line can pass through two distinct

points. So, the assumption that we started with, that two lines can pass through two

distinct points is wrong.

     .A                              B

Summary:-

1. Though Euclid defined a point, a line, and a plane, the definitions are not accepted by

mathematicians. Therefore, these terms are now taken as undefined.

2. Axioms or postulates are the assumptions which are obvious universal truths. They are not

proved.

3. Theorems are statements which are proved, using definitions, axioms, previously proved

statements and deductive reasoning.

4. Some of Euclid’s axioms were :

(1) Things which are equal to the same thing are equal to one another.

(2) If equals are added to equals, the wholes are equal.

(3) If equals are subtracted from equals, the remainders are equal.

(4) Things which coincide with one another are equal to one another.

(5) The whole is greater than the part.

(6) Things which are double of the same things are equal to one another.

(7) Things which are halves of the same things are equal to one another.

5. Euclid’s postulates were :

Postulate 1 : A straight line may be drawn from any one point to any other point.

Postulate 2 : A terminated line can be produced indefinitely.

Postulate 3 : A circle can be drawn with any centre and any radius.

Postulate 4 : All right angles are equal to one another.

Postulate 5 : If a straight line falling on two straight lines makes the interior angles on the

same side of it taken together less than two right angles, then the two straight lines, if

produced indefinitely, meet on that side on which the sum of angles is less than two right

angles.

6. Two equivalent versions of Euclid’s fifth postulate are:

(i) ‘For every line l and for every point P not lying on l, there exists a unique line m

passing through P and parallel to l’.

(ii) Two distinct intersecting lines cannot be parallel to the same line.

7. All the attempts to prove Euclid’s fifth postulate using the first 4 postulates failed. But they

led to the discovery of several other geometries, called non-Euclidean geometries.

Chapter 4.

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