NCERT Class 9 Solutions: Introduction to Euclid՚s Geometry (Chapter 5) Exercise 5.2 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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?

Solution:

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.

Solution:

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.

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