.. index:: proof, triangles, angles

.. _I.17:
.. _sum of any two angles of triangle is less than two right angles:

the sum of any two angles of a triangle is less than two right angles
=====================================================================

  I.17

  In any triangle two angles taken together in any manner [1]_ are less
  than two right angles.

  -- Euclid


.. image:: elem.1.prop.17.png
   :align: right
   :width: 300px

Let `ABC` be a triangle; I say that two angles of the triangle `ABC` taken together in any manner are less than two right angles.

For let `BC` be produced to `D`. [I.post.2]

Then, since the angle `ACD` is an exterior angle of the triangle `ABC`,

it is greater than the interior and opposite angle `ABC`. [I.16] <pb n="282"/>

- Let the angle `ACB` be added to each;

therefore the angles `ACD`, `ACB` are greater than the angles `ABC`, `BCA`.  But the angles `ACD`, `ACB` are equal to two right angles. [I.13]

Therefore the angles `ABC`, `BCA` are less than two right angles.

Similarly we can prove that the angles `BAC`, `ACB` are also less than two right angles, and so are the angles `CAB`, `ABC` as well.

Therefore etc.

-  Q. E. D.

references
----------


[I.13]: /elem.1.13 "Book 1 - Proposition 13"
[I.16]: /elem.1.16 "Book 1 - Proposition 16"
[I.post.2]: /elem.1.post.2 "Book 1 - Postulate 2"

footnotes
---------


.. [1] taken together in any manner,
    <foreign lang="greek">πάντη μεταλαμβανόμεναι</foreign>, i.e. any pair added together.