.. index:: proof, parallels

.. _I.30:
.. _parallels are parallel to each other:

parallels are parallel to each other
====================================

  I.30
  
  Straight lines parallel to the same straight line are also parallel to one another.

  -- Euclid


.. image:: elem.1.prop.30.png
   :align: right
   :width: 300px

Let each of the straight lines `AB`, `CD` be parallel to `EF`; I say that `AB` is also parallel to `CD`. 

For let the straight line `GK` fall upon them; 

Then, since the straight line `GK` has fallen on the parallel straight lines `AB`, `EF`, 

- the angle `AGK` is equal to the angle `GHF`. [I.29]

Again, since the straight line `GK` has fallen on the parallel straight lines `EF`, `CD`, 

- the angle `GHF` is equal to the angle `GKD`. [I.29]

But the angle `AGK` was also proved equal to the angle `GHF`; 

- therefore the angle `AGK` is also equal to the angle `GKD`; [I.c.n.1]

and they are alternate. 

Therefore `AB` is parallel to `CD`. [1]_

- Q. E. D.

references
----------

[I.29]: /elem.1.29 "Book 1 - Proposition 29"
[I.c.n.1]: /elem.1.c.n.1 "Book 1 - Common Notion 1"

footnotes
---------

.. [1] Therefore...

   The usual <em>conclusion</em> in general terms (<quote>Therefore
   etc.</quote>) repeating the enunciation is, curiously enough, wanting at the
   end of this proposition.