NCERT Class 9 Maths Solutions Chapter - 5 Introduction to Euclid's Geometry, Ex - 5.2

Ex - 5.2

Question 1.  How would you rewrite Euclid's fifth postulate so that it would be easier to understand?

Solution

Two lines are said to be parallel if they are equidistant from one other and they do not have any point of intersection. In order to understand it easily let us take any line l and a point P not on l. Then, by Play fair's axiom (equivalent to the fifth postulate), we know that there is a unique line m through P which is parallel to l.

 

                               

 

Now, the distance of a point from a line is the length of the perpendicular from the point to the line. Let AB be the distance of any point on m from l and CD be the distance of any point on l from m. We can observe that AB = CD. In this way the distance will be the same for any point on m from l and any point on l from m. So, these two lines are everywhere equidistant from one another.

 

                                        

Question 2.  Does Euclid's fifth postulate imply the existence of parallel lines? Explain.

Solution

Yes             
    According to Euclid's 5^th postulates when n line falls on l and m and if 
 , then producing line l and m further will meet in

 

the side of 1 and 2 which is less than 180�

                        

 

If ,

 

 

                         

 

 

Now, the lines l and m neither meet at the side of 1 and 2 nor at the side of 3 and 4. That means the lines l and m will never intersect each other. Therefore, we can say that the lines are parallel.