NCERT Class 9 Solutions: Introduction to Euclid՚s Geometry (Chapter 5) Exercise 5.2 (For CBSE, ICSE, IAS, NET, NRA 2023)

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Euclid՚s 5th postulates

Euclid՚s fifth postulates states that If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side

Q-1 How would you rewrite Euclid՚s fifth postulate so that it would be easier to understand?


First lets understand the Euclid՚s 5th postulate. It is equivalent to what is known as the parallel postulate. In the below figure , thus the lines intersect on that side.

Lines Intersect Because Angle Alpha + Angle Beta is Less Tha …

On the other hand:

Since Angle a + Angle B is 180 Therefore Lines Are Parallel

The two angles a and b are known as consecutive interior angles. If the sum of these consecutive interior angles is , then the lines do not intersect i.e.. they are parallel. Thus Euclid՚s 5th postulate can be written as “When two parallel lines intersect a third line, then the consecutive interior angles are supplementary.”

Q-2 Does Euclid՚s fifth postulate implies the existence of parallel lines? Explain.


Euclids 5th Postulate and Parallel Lines

As we understood in the previous question, when the two lines (m and n above) intersect the third line (l) . Then, if the sum of angle 4 and 3 above is less than , then lines will intersect on side A. If the sum of angle 1 and 2 above is less than , then lines will intersect on side B. If both and , than lines m and n will be parallel and will not intersect on either side.