The straight lines joining equal and parallel straight lines (at the extremities which are) in the same directions (respectively) are themselves also equal and parallel.
Let AB, CD
be equal and parallel, and let the straight
lines AC, BD
join them (at the extremities which are) in the same directions
I say that AC, BD are also equal and parallel.
Let BC be joined.
Then, since AB is parallel to
BC has fallen
the alternate angles ABC, BCD are equal to one another. [I. 29]
And, since AB is equal to
and BC is common,
the two sides AB, BC are equal to the two sides DC, CB;
and the angle ABC is equal to the angle BCD;
therefore the base AC is equal to the base BD,
and the triangle ABC is equal to the triangle DCB,
and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend; [I. 4]
therefore the angle ACB is equal to the angle CBD.
And, since the straight line BC
falling on the two straight lines
has made the alternate angles equal to one another,
AC is parallel to BD. [I. 27]
And it was proved equal to it.
Therefore etc. Q.E.D.
Book I: Euclid, Book I (ed. Sir Thomas L. Heath 1st Edition, 1908)
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