GSEB Class 9 Maths Solutions Chapter 5 Introduction to Euclids Geometry Exercise 5.2

Question 1. How would you rewrite Euclid's fifth postulate so that it would be easier to understand?
Answer: We say two lines are parallel if they stay the same distance apart and never meet. Imagine a line \(l\) and a point \(P\) that is not on line \(l\). According to Playfair's axiom, which is like Euclid's fifth postulate, only one unique line \(m\) can go through point \(P\) and be parallel to line \(l\). The gap between line \(l\) and line \(m\) is shown by \(PQ\), which is a straight line drawn from point \(P\) to line \(l\) at a 90-degree angle. Also, if we draw another perpendicular line \(RS\) from any point \(R\) on line \(m\) to line \(l\), then \(PQ\) will be equal to \(RS\). This shows that the distance between these two lines is always the same, meaning they remain equally far apart everywhere.
In simple words: Two lines are parallel if they always stay the same distance apart and never cross. Euclid's fifth postulate helps us understand this idea by showing that only one line can pass through a point and be parallel to another line.

Exam Tip: When explaining postulates, it helps to use simple language and relate them to real-world concepts like railway tracks for parallel lines.

Question 2. Does Euclid's fifth postulate imply the existence of parallel lines? Explain.
Answer: Yes, Euclid's fifth postulate implies the existence of parallel lines. Based on the fifth postulate, if a line \(n\) cuts across two other lines \(l\) and \(m\) at different points (and these lines are not parallel), then the total of the two inside angles on one side will always be less than 180 degrees. This means if \( \angle 1 + \angle 2 < 180^\circ \), then lines \(l\) and \(m\) will eventually cross each other on that side. Conversely, if the sum of these interior angles on one side is exactly 180 degrees (for instance, \( \angle 1 + \angle 2 = 180^\circ \) or \( \angle 3 + \angle 4 = 180^\circ \)), then the lines \(l\) and \(m\) are parallel and will never intersect. This condition directly describes parallel lines.
In simple words: Yes, the fifth postulate shows how parallel lines work. If two lines are cut by another line, and the angles inside on one side add up to exactly 180 degrees, then those two lines will never meet, meaning they are parallel.

Exam Tip: Clearly state "Yes" or "No" first, then use the conditions of Euclid's fifth postulate (sum of interior angles on the same side) to explain the concept of parallel lines.