 Hello and welcome to the session. In this session we will discuss Euclid's postulates. Now first let's see what are postulates. Postulates basically are the assumptions which are obvious universal truths. They are not proved. Say the postulates and axioms are the same but there is a difference between them as axioms are used throughout mathematics. The postulates are linked only to geometry. Now let's define theorems. Theorems are basically the statements which are proved using definitions, previously proved statements and deductive reasoning. Now we have five Euclid's postulates. The first postulate is the straight line may be drawn from any one point to any other point. We can state this result in the form of an axiom also according to which we have that given two distinct points there is a unique line that passes through them. Now postulate two at a terminated line and be produced indefinitely. Postulate three says a circle can be drawn with any center, any radius. The next postulate that is postulate four says all right angles are equal to one another. Next we have postulate five according to which we have that 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. Postulate the line PQ falls on the lines AB and CD such that the sum of the interior angles that is angle one plus angle two is less than 180 degrees on the left side of PQ then the lines AB and CD will intersect on the left side of PQ. This is Euclid's fifth postulate. Now we have an important theorem which says that two distinct lines cannot have more than one point in common. This theorem would be used frequently in different results. Now we discuss equivalent versions of Euclid's fifth postulate. Euclid's fifth postulate is very significant in the history of mathematics. There are several equivalent versions of this postulate. One of them is play fairs axiom according to which we have that for every line L for every point P not lying on L there exists a unique line passing through P and parallel to L. Consider this line L and a point P which is not lying on this line L. Then as you can see there exists a unique line this M which is passing through this point P and this line M is parallel to L and the other form or the other version of Euclid's fifth postulate can also be written as two distinct intersecting lines cannot be parallel to the same line. All the attempts to prove Euclid's fifth postulate using the first four postulates failed but they led to the discovery of several other geometries called non-Euclidean geometries. Consider spherical geometry which is non-Euclidean geometry and it is basically geometry of the universe we live in. Then in spherical geometry we have that the lines are not straight they are basically parts of great circles and thus we say that Euclidean geometry is valid only for the figures in the plane on the curved surfaces it fails. So this completes the session hope you have understood Euclid's fifth postulates.