.. index:: proof, angles, lines

.. _I.14:
.. _adjacent angles equal to two right angles make a line:

adjacent angles equal to two right angles make a line
=====================================================

  I.14

  If with any straight line, and at a point on it, two straight lines not lying
  on the same side make the adjacent angles equal to two right angles, the two
  straight lines will be in a straight line with one another. [1]_

  -- Euclid


.. image:: elem.1.prop.14.png
   :align: right
   :width: 300px

For with any straight line `AB`, and at the point `B` on it, let the two straight lines `BC`, `BD` not lying on the same side make the adjacent angles `ABC`, `ABD` equal to two right angles;

I say that `BD` is in a straight line with `CB`.

For, if `BD` is not in a straight line with `BC`, let `BE` be in a straight line with `CB`.

Then, since the straight line `AB` stands on the straight line `CBE`,

- the angles `ABC`, `ABE` are equal to two right angles. [<a href="/elem.1.13">I. 13</a>]

But the angles `ABC`, `ABD` are also equal to two right angles;

- therefore the angles `CBA`, `ABE` are equal to the angles `CBA`, `ABD`. [<a href="/elem.1.post.4">Post. 4</a> and <a href="/elem.1.c.n.1">C.N. 1</a>] [2]_

Let the angle `CBA` be subtracted from each; therefore the remaining angle `ABE` is equal to the remaining angle `ABD`, [<a href="/elem.1.c.n.3">C.N. 3</a>]

- the less to the greater: which is impossible. Therefore `BE` is not in a straight line with `CB`.

Similarly we can prove [3]_ that neither is any other straight line except `BD`. <pb n="277"/>

- Therefore `CB` is in a straight line with `BD`.

Therefore etc.

- Q. E. D.

.. [1] If with any straight line....
    There is no greater difficulty in translating the works of the Greek geometers than that of accurately giving the force of prepositions. <foreign lang="greek">πρός</foreign>, for instance, is used in all sorts of expressions with various shades of meaning. The present enunciation begins <foreign lang="greek">ἐὰν πρός τινι εὐθείᾳ καὶ τῷ πρὸς αὐτῆ σημείῳ</foreign>, and it is really necessary in this one sentence to translate <foreign lang="greek">πρός</foreign> by three different words, `with`, <em>at</em>, and <em>on</em>. The first <foreign lang="greek">πρός</foreign> must be translated by <em>with</em> because two straight lines <quote>make</quote> an angle <em>with</em> one another. On the other hand, where the similar expression <foreign lang="greek">πρὸς τῇ δοθείση εὐθείᾳ</foreign> occurs in <a href="/elem.1.23">I. 23</a>, but it is a question of <quote>constructing</quote> an angle (<foreign lang="greek">συστἡσασθαι</foreign>), we have to say <quote>to construct <em>on</em> a given straight line.</quote> <title>Against</title> would perhaps be the English word coming nearest to expressing all these meanings of <foreign lang="greek">πρός</foreign>, but it would be intolerable as a translation.

.. [2] equal to the angles
    Todhunter points out that for the inference in this line <a href="/elem.1.post.4">Post. 4</a>, that all right angles are equal, is necessary as well as the Common Notion that things which are equal to the same thing (or rather, here, to <em>equal things</em>) are equal. A similar remark applies to steps in the proofs of <a href="/elem.1.15">I. 15</a> and <a href="/elem.1.28">I. 28</a>.

.. [3] we can prove.
    The Greek expresses this by the future of the verb, <foreign lang="greek">δείξομεν</foreign>, <quote>we shall prove,</quote> which however would perhaps be misleading in English.