.. _I.15:
.. _opposing angles of intersecting lines are equal:

opposing angles of intersecting lines are equal
===============================================

.. index:: proof, angles, lines

.. image:: elem.1.prop.15.png
   :align: right
   :width: 300px

..

  I.15

  If two straight lines cut one another, they make the vertical angles [1]_ equal to one another.

  -- Euclid


For let the straight lines `AB`, `CD` cut one another at the point `E`; 

I say that the angle `AEC` is equal to the angle `DEB`, 

- and the angle `CEB` to the angle `AED`.

For, since the straight line `AE` stands on the straight line `CD`, making the angles `CEA`, `AED`, 

- the angles `CEA`, `AED` are equal to two right angles [I.13]

Again, since the straight line `DE` stands on the straight line `AB`, making the angles `AED`, `DEB`, 

- the angles `AED`, `DEB` are equal to two right angles. [I.13]

But the angles `CEA`, `AED` were also proved equal to two right angles; 

- therefore the angles `CEA`, `AED` are equal to the angles `AED`
`DEB`. [I.post.4] and [I.cn.1] Let the angle `AED` be subtracted from each; therefore the remaining angle `CEA` is equal to the remaining angle `BED`. [I.cn.3]

Similarly it can be proved that the angles `CEB`, `DEA` are also equal.

Therefore etc. 

- Q. E. D. 

porism
------

[From this it is manifest that, if two straight lines cut one another, they will make the angles at the point of section [2]_ equal to four right angles.

references
----------

[I.13]: /I.13 "Book I - Proposition 13"
[I.post.4]: /I.post.4 "Book I - Postulate 4"
[I.cn.1]: /I.cn.1 "Book I - Common Notion 1"
[I.cn.3]: /I.cn.3 "Book I - Common Notion 3"

footnotes
---------

.. [1] the vertical angles.
    The difference between <em>adjacent</em> angles (<foreign lang="greek">αἱ ἐφεξῆς γωνίαι</foreign>) and <em>vertical</em> angles (<foreign lang="greek">αἱ κατὰ κορυφὴν γωνίαι</foreign>) is thus explained by Proclus (<xref n="Proc. p. 298, 14-24" from="ROOT" to="DITTO">p. 298, 14-24</xref>). The first term describes the angles made by two straight lines when one only is divded by the other, i.e. when one straight line meets another at a point which is not either of its extremities, but is not itself produced beyond the point of meeting. When the first straight line <em>is</em> produced, so that the lines cross at the point, they make two pairs of <em>vertical</em> angles (which are more clearly described as <em>vertically opposite</em> angles), and which are so called because their convergence is from opposite directions to one point (the intersection of the lines) as vertex (<foreign lang="greek">κορυφή</foreign>).

.. [2] at the point of section,
    literally <quote>at the section,</quote> <foreign lang="greek">πρὸς τῇ τομῆ</foreign>.