 Hi and welcome to the session. I am Harsha and I am going to help you with the following question that says, does Euclid's fifth postulate imply the existence of parallel lines? Explain. So let us now begin with the solution and the answer is yes, Euclid's fifth postulate implies the existence of parallel lines and let us say how if a line L falls on two straight lines M and N such that the sum of two interior angles on one side of L, let us say angle 1, let us say angle 2 such that angle 1 plus angle 2 is equal to two right angles then the Euclid's fifth postulate the lines will not meet on this side of L that is L and M will never intersect on the right hand side of this L. Next we know that the sum of interior angles on the other side would also be two right angles and again this will imply that if this is angle 3 and this is angle 4 at angle 3 plus angle 4 will also be equal to two right angles and again the Euclid's fifth postulate these two lines will not intersect on the left hand side of L and thus we can say that the two lines will not meet and hence we can say that the two lines are parallel. Thus our answer is if a straight line L falls on two straight lines M and N such that the sum of the interior angles on one side of L is two right angles then by Euclid's fifth postulate the lines will not meet on this side of L. Next you know that the sum of interior angles on the other side of L will also be two right angles therefore they will not meet on the other side also and thus we can say that the lines M and N will never meet and are therefore parallel. So this completes the solution hope you enjoyed it take care and bye for now.