Book 1 - Proposition 33

Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel.

Let AB and CD be equal and parallel, and let the straight lines AC and BD join them at their ends in the same directions.

I say that AC and BD are also equal and parallel.

Join BC.

Since AB is parallel to CD, and BC falls upon them, therefore the alternate angles angle ABC and angle BCD equal one another.

Since AB equals CD, and BC is common, the two sides AB and BC equal the two sides DC and CB, and the angle ABC equals the angle BCD, therefore the base AC equals the base BD, the triangle ABC equals the triangle DCB, and the remaining angles equals the remaining angles respectively, namely those opposite the equal sides. Therefore the angle ACB equals the angle CBD.

Since the straight BC falling on the two straight lines AC and BD makes the alternate angles equal to one another, therefore AC is parallel to BD.

And it was also proved equal to it.

Therefore straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel.
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