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Does Euclid’s fifth postulate imply the existence of parallel lines? Explain. - Mathematics

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ConceptEquivalent Versions of Euclid’S Fifth Postulate

Question

Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.

Solution

Yes.

According to Euclid’s 5th postulate, when n line falls on l and m and if

∠1 + ∠2 < 180°, then ∠3 + ∠4 > 180°, producing line l and m further will meet in the side of ∠1 and ∠2 which is less than 180°.

If ∠1 + ∠2 = 180°, then ∠3 + ∠4 = 180°

The lines l and m neither meet at the side of ∠1 and ∠2 nor at the side of ∠3 and ∠4. This means that the lines l and m will never intersect each other. Therefore, it can be said that the lines are parallel.

  Is there an error in this question or solution?

APPEARS IN

 NCERT Solution for Mathematics Class 9 (2018 to Current)
Chapter 5: Introduction to Euclid's Geometry
Ex. 5.2 | Q: 2 | Page no. 88

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Solution Does Euclid’s fifth postulate imply the existence of parallel lines? Explain. Concept: Equivalent Versions of Euclid’S Fifth Postulate.
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