NCERT Class 9 Solutions: Introduction To Euclid's Geometry (Chapter 5) Exercise 5.2

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 α+β<180° , thus the lines intersect on that side.

Lines intersect because angle alpha + angle beta is less than 180 degree

Parallel Postulate

Lines intersect because angle alpha + angle beta is less than 180 degree

On the other hand:

Since angle a + angle b is 180 therefore lines are parallel.

Angle a + Angle B = 180 Degrees

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 180° , 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

Parallel Lines With Euclid's Postulate

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 180° , then lines will intersect on side A. If the sum of angle 1 and 2 above is less than 180° , then lines will intersect on side B. If both 1+2=180° and 3+4=180° , than lines m and n will be parallel and will not intersect on either side.

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