Theorem. If Euclid's 5th postulate holds, Playfair's axiom is true.

Proof. Euclid's postulate states that given two line segments $a$ and $b$ and given their transversal $c$, $a$ and $b$'s extensions will meet, if their interior angles on the same side of $c$ sum up to form less than two right angles.

Playfair's axiom says that given a straight line $a$ and a point not on this line, there's a unique line passing through that point which is parallel to $a$.

Suppose Playfair is not right, then

(i) either all lines passing through that point are not parallel to $a$
or
(ii) there are at least two which are parallel to $a$.

If (i) was right then there is a pair of straight lines such that their interior angles add up to two right angles and they'll meet.
If (ii) was right then there is a pair of straight lines such that their interior angles don't add up to two right angles and they won't meet.

If (i) holds the proposition of Euclid's 5th is false, resulting in the implication $$ \neg( \angle_i (a,c) + \angle_i (b,c) < \pi \rightarrow a,b \: \nparallel ) \leftrightarrow \underbrace{ \underbrace{\angle_i (a,c) + \angle_i (b,c) < \pi}_{\text{false}} \rightarrow \underbrace{\neg(a,b \: \nparallel)}_{\text{false}} }_{\text{true}} $$ to be true.

If (ii) holds the proposition of Euclid's 5th is true, resulting in the implication $$ \angle_i (a,c) + \angle_i (b,c) < \pi \rightarrow \neg(a,b \: \nparallel) $$ to be true.

Hence, in every case the negation of Playfair's axiom is true the negation of Euclid's 5th is true. $\Box$

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2014 DEC 27 (v1.0)
Contact me: m.herrmann followed by an -at- followed by blaetterundsterne.org.

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