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Euclid’s Postulates


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

According to this postulate at least one straight line passes through two distinct points, but it does not say that there cannot be more than one such line. In his work Euclid has frequently assumed, without mentioning, that there is a unique line joining two points.

Axiom 5.1 : Given two distinct point, there is a unique line that passes through them


In the above figure, the only line which passes through P as well as Q is the line PQ

Postulate 2 : A terminated line can be produced indefinitely.

According to present days’ terms, this postulate tells us that a line segment can be extended on either side to form a line.

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; the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

Of these five postulates, postulates 1 to 4 are simple and obvious that these are taken as self evident truths. Postulate 5 is more complex than other postulate. There are several equivalent versions of this postulate. One of them is ‘Playfair Axiom’ which was given by a Scottish mathematician Johan Playfair, given below:

‘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

From the diagram, we can see that of all the lines passing through the point P, only line m is parallel to line l.

We can also state this result in the following form :

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

Theorem 5.1:

  Two distinct lines cannot have more than one point in common.

Here we are given two lines
λ 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.


What can we conclude?


We are forced to conclude that two distinct lines cannot have more than one point in common.



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