.. index:: proof, parallels

.. _I.33:
.. _similar angles on parallels:

similar angles on parallels
===========================

  I.33

  The straight lines joining equal and parallel straight lines (at the
  extremities which are) in the same directions (respectively) [1]_  are
  themselves also equal and parallel.

  -- Euclid

.. image:: elem.1.prop.33.png
   :align: right
   :width: 300px

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 (respectively); I say that `AC`, `BD` are also equal and parallel.

Let `BC` be joined.

Then, since `AB` is parallel to `CD`, and `BC` 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`; [2]_  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 `AC`, `BD` has made the alternate angles equal to one another,

- `AC` is parallel to `BD`. [I.27]

And it was also proved equal to it.

Therefore etc.

- Q. E. D.

references
----------

[I.4]: /elem.1.4 "Book 1 - Proposition 4"
[I.27]: /elem.1.27 "Book 1 - Proposition 27"
[I.29]: /elem.1.29 "Book 1 - Proposition 29"

footnotes
---------

.. [1] joining...(at the extremities which are) in the same directions (respectively).

   I have for clearness' sake inserted the words in brackets though they are
   not in the original Greek, which has <quote>joining...in the same
   directions</quote> or <quote>on the same sides,</quote> <foreign
   lang="greek">ἐπὶ τὰ αυτὰ μέρη ἐπιζευγνύουσαι</foreign>. The expression
   <quote>tiwards the same parts,</quote> though usage has sanctioned it, is
   perhaps not quite satisfactory.

.. [2] DC, CB

   and 18. `DCB`. The Greek has <quote> `BC`, `CD`</quote> and
   <quote>`BCD`</quote> in these places respectively. Euclid is not always
   careful to write in corresponding order the letters denoting corresponding
   points in congruent figures. On the contrary, he evidently prefers the
   alphabetical order, and seems to disdain to alter it for the sake of
   beginners or others who might be confused by it. In the case of angles
   alteration is perhaps unnecessary; but in the case of triangles and pairs of
   corresponding sides I have ventured to alter the order to that which the
   mathematician of to-day expects.