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.
be equal and parallel, and let the straight
join them (at the extremities which are) in the same directions (respectively); I say that AC
are also equal and parallel.
Then, since AB
is parallel to CD
has fallen upon them, the alternate angles ABC, BCD are equal to one another. [I. 29]
And, since AB
is equal to CD
, 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 griangle 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 AC
has made the alternate angles equal to one another, AC is parallel to BD. [I. 27]
And it was also proved equal to it.
Q. E. D.