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Statement: Prove, under the assumption of the parallel postulate (P-1), parallelism of lines is transitive. That is if l||m and m||q, then l||q.

Parallel Postulate(p-1)-If l is any line and point P not on l there exists an unique line passing through P parallel to l( in the plane of P,l).

Proof- Assume to the contrary that l is not parallel to q. Further assume the parallel postulate p-1. Sine l is not parallel to q that means both lines meet at least 1 point.But that's a contradiction since it contradicts parallel postulate p-1.

Is that correct? or do i need to explain it a bit more why it contradicts?

user60887
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  • You should explain why it's a contradiction. – Jim Feb 22 '13 at 16:42
  • I would explain more, maybe by taking $P$ to be the intersection of $l$ and $q$ and chasing down what the parallel postulate gives you. – Louis Feb 22 '13 at 16:44

2 Answers2

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Hypothesis: $ \ell \parallel m $ and $ m \parallel q $.

  • Suppose that $ \ell = q $. By convention, $ \ell \parallel q $.

  • Suppose that $ \ell \neq q $. By way of contradiction, assume that $ \ell \not\parallel q $. Then $ \ell $ and $ q $ intersect in at least one point $ x $, which implies that $ \ell $ and $ q $ are distinct lines parallel to $ m $ passing through $ x $. We thus contradict the Parallel Postulate that there exists only one line parallel to $ m $ passing through $ x $. Our assumption that $ \ell \not\parallel q $ is therefore false, so we conclude that $ \ell \parallel q $.

Note: In order to derive a contradiction, you need to explicitly assume that $ \ell \neq q $.

Haskell Curry
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Having graded for a geometry class before I'll tell you that I would take a few points off for that answer.

Were I your grader I would like you to say explicitly why $P$ doesn't lie on $m$, that the parallel postulate tells you there is exactly one line through $P$ parallel to $m$, and emphasize that $l$ and $q$ are distinct lines both parallel to $m$ (and what to do if they are not distinct!).

The idea behind your proof is completely correct, all you need now is to be extra careful as you explain it.

Jim
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