Question 11
Question: Euclidean geometry is valid only for the figures in the plane. Explain.
Answer:
According to Euclidean geometry, lines are straight. But according to spherical geometry, lines are not straight. They are part of large circles. The lines AN and BC are perpendicular to the same line AB. Postulate 5 fails, because the lines BN and AN intersect at N, though the sum of the angles on the same side of the line AB is not less than 180o.
Moreover since 
, the sum of the triangle NAB is grater 180o. Thus, Euclidean geometry is valid only for the figures in the plane on the curved surface it fails.
Question 12
Question: How would you rewrite fifth postulate so that it is easier to understand?
Answer: Suppose line l falls on two given lines m and n such that the sum of the interior angles on the same side of l, is 180o. Then the two lines do not intersect, whatever they are produced.
Let us think what would happen if sum of the interior angles on the same side are greater than 180o. In this case eventually the sum of the interior angles on the other side will be less than 180o. Therefore they intersect on the other side of l.
Question 13
Question: Does Euclid's fifth postulate imply the existence of parallel lines.
Answer: Yes, Euclid's fifth postulate imply the existence of parallel lines. If a straight line l falls on two straight lines m and n such that the sum of interior angles on one side of l is two right angles, then according to Euclid's fifth postulate they do not meet on this side of l.
Since the sum of the interior angles on the other side also will be two right angles, the lines m and n do not intersect on other side also.
Therefore the lines m and n do not intersect on either side. Therefore they are parallel to each other.
Question 14
Question: Consider the following statement: There exist a pair of straight lines that are everywhere equidistant from one another. Is the statement a direct consequence of Euclid's fifth postulate? Explain.
Answer: Take any line l and a point p on l, then we know that then is a unique line m through p which is parallel to l.
We also know that the distance of a point from a line is its perpendicular length.
Take any point on m and draw a perpendicular on l. Let the length of the perpendicular be d1.
Take any point on l and draw a perpendicular on m. Let the length of the perpendicular be d2.
It will be seen that d1 = d2 These two parallel lines are equidistant from one another.
