Euclid, Elements of Geometry, Book I, Proposition 33
(Edited by Dionysius Lardner, 1855)

Proposition XXXIII. Theorem.
[Euclid, ed. Lardner, 1855, on Google Books]

(150) Right lines (A C and B D) which join the adjacent extremities of two equal and parallel right lines (A B and C D) are themselves equal and parallel.

Draw the diagonal A D, and in the triangles C D A and B A D the sides C D and B A are equal (by hyp); A B C D A D is common to both triangles, and the angle C D A is equal to the alternate B A D (XXIX); therefore the lines A C and B D are equal, and also the angles C A D and B D A; therefore the right line A D cutting the right lines A C and B D makes the alternate angles equal, and therefore (XXVII) the right lines A C and B D are parallel.

Book I: Euclid, Book I (ed. Dionysius Lardner, 11th Edition, 1855)

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