.. _I.16:
.. _exterior angle of triangle is greater than either the interior or opposite angles:

exterior angle of a triangle is greater than the either the interior or opposite angles
=======================================================================================

.. index:: proof, angles, triangles

.. image:: elem.1.prop.16.png
   :align: right
   :width: 300px

..

  I.16

  In any triangle, if one of the sides be produced, the exterior angle is
  greater than either of the interior and opposite angles.

  -- Euclid


Let `ABC` be a triangle, and let one side of it `BC` be produced to `D`; 

I say that the exterior angle [1]_ `ACD`is greater than either of the interior and opposite angles [2]_ `CBA`, `BAC`.

Let `AC` be bisected at `E` [I.10], and let `BE` be joined and produced in a straight line to `F`;

let `EF` be made equal to `BE`[I.3], let `FC` be joined [I.post.1], and let `AC` be drawn through to `G` [3]_ [I.post.2].

Then, since `AE` is equal to `EC`, and `BE` to `EF`, 

- the two sides `AE`, `EB` are equal to the two sides `CE`, `EF` respectively; and the angle `AEB` is equal to the angle `FEC`, for they are vertical angles. [I.15] Therefore the base `AB` is equal to the base `FC`, and the triangle `ABE` is equal to the triangle `CFE` [4]_, and the remaining angles are equal to the remaining angles respectively, namely those which the equal sides subtend; [I.4] therefore the angle `BAE` is equal to the angle `ECF`.

But the angle `ECD` is greater than the angle `ECF`; [I.c.n.5] 

- therefore the angle `ACD` is greater than the angle `BAE`.

Similarly also, if `BC` be bisected, the angle `BCG`, that is, the angle `ACD` [I.15], can be proved greater than the angle `ABC` as well.

Therefore etc.

- Q. E. D.

references
----------


[I.3]: /elem.1.3 "Book 1 - Proposition 3"
[I.4]: /elem.1.4 "Book 1 - Proposition 4"
[I.10]: /elem.1.10 "Book 1 - Proposition 10"
[I.15]: /elem.1.15 "Book 1 - Proposition 15"
[I.post.1]: /elem.1.post.1 "Book 1 - Postulate 1"
[I.post.2]: /elem.1.post.2 "Book 1 - Postulate 2"
[I.c.n.5]: /elem.1.c.n.5 "Book 1 - Common Notion 5"

footnotes
---------


.. [1] the exterior angle,
    literally <quote>the outside angle,</quote> <foreign lang="greek">ἡ ἐκτὸς γωνία</foreign>.

.. [2] the interior and opposite angles,
    <foreign lang="greek">τῶν ἐντὸς καὶ ἀπεναντίον γωνιῶν</foreign>.

.. [3] let AC be drawn through to G.
    The word is <foreign lang="greek">διήχθω</foreign>, a variation on the more usual <foreign lang="greek">ἐκβεβλήσθω</foreign>, <quote>let it be <em>produced</em>.</quote>

.. [4] CFE,
    in the text <quote>`FEC`.</quote>