# Proof: By Euclid

(related to Proposition: 1.27: Parallel Lines I)

• For if not, being produced, $AB$ and $CD$ will certainly meet together: either in the direction of $B$ and $D$, or (in the direction) of $A$ and $C$ [Def. 1.23] .
• Let them have been produced, and let them meet together in the direction of $B$ and $D$ at (point) $G$.
• So, for the triangle $GEF$, the external angle $AEF$ is equal to the interior and opposite (angle) $EFG$.
• The very thing is impossible [Prop. 1.16].
• Thus, being produced, $AB$ and $CD$ will not meet together in the direction of $B$ and $D$.
• Similarly, it can be shown that neither (will they meet together) in (the direction of) $A$ and $C$.
• But (straight lines) meeting in neither direction are parallel [Def. 1.23] .
• Thus, $AB$ and $CD$ are parallel.
• Thus, if a straight line falling across two straight lines makes the alternate angles equal to one another then the (two) straight lines will be parallel (to one another).
• (Which is) the very thing it was required to show.

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