.. index:: proof, parallels

.. _I.29:
.. _parallel angles:

parallel angles
===============

  I.19

  A straight line falling on parallel straight lines makes the alternate angles
  equal to one another, the exterior angle equal to the interior and opposite
  angle, and the interior angles on the same side equal to two right angles. 

  -- Euclid

.. image:: elem.1.prop.29.png
   :align: right
   :width: 300px

For let the straight line `EF` fall on the parallel straight lines `AB`, `CD`;

I say that it makes the alternate angles `AGH`, `GHD` equal, the exterior angle `EGB` equal to the interior and opposite angle `GHD`, and the interior angles on the same side, namely `BGH`, `GHD`, equal to two right angles.

For, if the angle `AGH` is unequal to the angle `GHD`, one of them is greater.

Let the angle `AGH` be greater. 

Let the angle `BGH` be added to each; therefore the angles `AGH`, `BGH` are greater than the angles `BGH`, `GHD`.

But the angles `AGH`, `BGH` are equal to two right angles; [I.13] 

- therefore the angles `BGH`, `GHD` are less than two right angles.

But straight lines produced indefinitely from angles less than two right angles meet; [I.post.5] [1]_ 

- therefore `AB`, `CD`, if produced indefinitely, will meet; but they do not meet, because they are by hypothesis parallel. [2]_ 

Therefore the angle `AGH` is not unequal to the angle `GHD`, 

- and is therefore equal to it.

Again, the angle `AGH` is equal to the angle `EGB`; [I.15] 

- therefore the angle `EGB` is also equal to the angle `GHD`. [I.c.n.1]

Let the angle `BGH` be added to each; 

- therefore the angles `EGB`, `BGH` are equal to the angles `BGH`, `GHD`. [I.c.n.2]

But the angles `EGB`, `BGH` are equal to two right angles; [I.13] 

- therefore the angles `BGH`, `GHD` are also equal to two right angles.

Therefore etc.

- Q. E. D.

references
----------

[I.13]: /elem.1.13 "Book 1 - Proposition 13"
[I.15]: /elem.1.15 "Book 1 - Proposition 15"
[I.post.5]: /elem.1.post.5 "Book 1 - Postulate 5"
[I.c.n.1]: /elem.1.c.n.1 "Book 1 - Common Notion 1"
[I.c.n.2]: /elem.1.c.n.2 "Book 1 - Common Notion 2"

footnotes
---------

.. [1] straight lines produced indefinitely from angles less than two right angles,

   <foreign lang="greek">αἰ δὲ ἀπ: ὲλασσόνων ἢ δύο ὀρθῶν ἐκβαλλόμεναι εἰς
   ἄπειπον συμπίπτουσιν</foreign>, a variation from the more explicit language
   of <a href="/elem.1.post.5">Postulate 5</a>. A good deal is left to be
   understood, namely that the straight lines begin from points at which they
   meet a transversal, and make with it internal angles on the same side the
   sum of which is less than two right angles.

.. [2] because they are by hypothesis parallel,
   
   literally <quote>because they are supposed parallel,</quote> <foreign
   lang="greek">διὰ τὸ παραλλήλους αὐτὰς ὑποκεῖσθαι</foreign>.