.. index:: proof, triangle

.. _I.19:
.. _greater angle of triangle is subtended by greater side:

the greater angle of a triangle is subtended by the greater side
================================================================

  I.19

  In any triangle the greater angle is subtended by the greater side.

  -- Euclid


.. image:: elem.1.prop.19.png
   :align: right
   :width: 300px

Let `ABC` be a triangle having the angle `ABC` greater than the angle `BCA`;

I say that the side `AC` is also greater than the side `AB`.

For, if not, `AC` is either equal to `AB` or less.

Now `AC` is not equal to `AB`; for then the angle `ABC` would also have been equal to the angle `ACB`; [I.5] but it is not;

- therefore `AC` is not equal to `AB`.

Neither is `AC` less than `AB`, for then the angle `ABC` would also have been less than the angle `ACB`; [I.18] but it is not;

- therefore `AC` is not less than `AB`.

And it was proved that it is not equal either.

- Therefore `AC` is greater than `AB`.

Therefore etc.

- Q. E. D.

references
----------

[I.5]: /elem.1.5 "Book 1 - Proposition 5"
[I.18]: /elem.1.18 "Book 1 - Proposition 18"