.. index:: proof, angles, lines

.. _I.13:
.. _angles from intersecting lines equal two right angles:

angles from intersecting lines equal two right angles (on one side)
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  I.13

  If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles.

  -- Euclid


.. image:: elem.1.prop.13.png
   :align: right
   :width: 300px

For let any straight line `AB` set up on the straight line `CD` make the angles `CBA`, `ABD`;

I say that the angles `CBA`, `ABD` are either two right angles or equal to two right angles. 


Now, if the angle `CBA` is equal to the angle `ABD`, 

- they are two right angles. [<a href="/elem.1.def.10">Def. 10</a>]

But, if not, let `BE` be drawn from the point `B` at right angles to `CD`; [<a href="/elem.1.11">I. 11</a>] 

- therefore the angles `CBE`, `EBD` are two right angles.

Then, since the angle `CBE` is equal to the two angles `CBA`, `ABE`, 

- let the angle `EBD` be added to each; [1]_ therefore the angles `CBE`, `EBD` are equal to the three angles `CBA`, `ABE`, `EBD`. [<title>C. N</title>. 2]

Again, since the angle `DBA` is equal to the two angles `DBE`, `EBA`, 

- let the angle `ABC` be added to each; therefore the angles `DBA`. `ABC` are equal to the three angles `DBE`, `EBA`, `ABC`. [<title>C. N</title>. 2]

But the angles `CBE`, `EBD` were also proved equal to the same three angles; 

- and things which are equal to the same thing are also equal to one another; [<title>C. N</title>. 1] therefore the angles `CBE`, `EBD` are also equal to the angles `DBA`, `ABC`.

But the angles `CBE`, `EBD` are two right angles; 

- therefore the angles `DBA`, `ABC` are also equal to two right angles.

Therefore etc.

- Q. E. D.

footnotes
---------

.. [1] let the angle EBD be added to each,
    literally <quote>let the angle `EBD` be added (so as to be) common,</quote> <foreign lang="greek">κοινὴ προσκείσθω ἡ ὑπὸ ΕΒΔ</foreign>. Similarly <foreign lang="greek">κοινὴ ἀφηρήσθω</foreign> is used of subtracting a straight line or angle from each of two others. <quote>Let the common angle `EBD` be added</quote> is clearly an inaccurate translation, for the angle is not common before it is added, i.e. the <foreign lang="greek">κοινὴ</foreign> is proleptic. <quote>Let the common angle be <em>subtracted</em></quote> as a translation of <foreign lang="greek">κοινὴ ἀφηρήσθω</foreign> would be less unsatisfactory, it is true, but, as it is desirable to use corresponding words when translating the two expressions, it seems hopeless to attempt to keep the word <quote>common,</quote> and I have therefore said <quote>to each</quote> and <quote>from each</quote> simply.