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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.

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#### 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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