Does Euclid's fifth postulate imply the existence of parallel lines? Explain
Solution:
Postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.
Yes, if ‘a’ and ‘b’ are two straight lines which are intersected by another line ‘c’, and the sum of co-interior angles are equal to 180°, then a || b.
According to Euclid’s 5th postulate,
∠1 + ∠2 < 180 then ∠3 + ∠4 > 180
[The interior angles on the same side of two straight lines which are intersected by another line taken together are less than two right angles]
Producing the line ‘a’ and ‘b’ further will meet in the side of which is less than 180°.
If ∠1 + ∠2 = 180 then ∠3 + ∠4 = 180
The lines ‘a’ and ‘b’ do not meet in the side where the angle is lesser than 180°
Thus, they will never intersect each other. Hence the two lines are said to be parallel to each other i.e. a || b
☛ Check: NCERT Solutions Class 9 Maths Chapter 5
Video Solution:
Does Euclid's fifth postulate imply the existence of parallel lines? Explain
NCERT Solutions Class 9 Maths Chapter 5 Exercise 5.2 Question 2
Summary:
Yes, Euclid's fifth postulate implies the existence of parallel lines.
☛ Related Questions:
- In Question 4, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.
- In Fig. 5.10, if AC = BD, then prove that AB = CD.
- Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate.)
- How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?