.. index:: proof, triangles

.. _I.18:
.. _the greater side of a triangle subtends the greater angle:

the greater side of a triangle subtends the greater angle
=========================================================

  I.18

  In any triangle the greater side subtends the greater angle.

  -- Euclid

.. image:: elem.1.prop.18.png
   :align: right
   :width: 300px

For let `ABC` be a triangle having the side `AC` greater than `AB`;

.. image:: elem.1.prop.18.b.png
   :align: right
   :width: 300px

I say that the angle `ABC` is also greater than the angle `BCA`.

For, since `AC` is greater than `AB`, let `AD` be made equal to `AB` [I.3], and let `BD` bejoined.

Then, since the angle `ADB` is an exterior angle of the triangle `BCD`,

it is greater than the interior and opposite angle `DCB`. [I.16]

But the angle `ADB` is equal to the angle `ABD`, 

- since the side `AB` is equal to `AD`;
  
therefore the angle `ABD` is also greater than the angle `ACB`;

therefore the angle `ABC` is much greater than the angle `ACB`. [1]_

Therefore etc.

- Q. E. D.


## References

[I.3]: /elem.1.3 "Book 1 - Proposition 3"
[I.16]: /elem.1.16 "Book 1 - Proposition 16"

## Footnotes

.. [1] Number

    In the enunciation of this number we have <foreign
   lang="greek">ὑποτείνειν</foreign> (<quote>subtend</quote>) used with the
   simple accusative instead of the more usual <foreign
   lang="greek">ὑπό</foreign> with accusative. The latter construction is used
   in the enunciation of <a href="/elem.1.19">I. 19</a>, which otherwise only
   differs from that of <a href="/elem.1.18">I. 18</a> in the order of the
   words. The point to remember in order to distinguish the two is that the
   <foreign lang="la">datum</foreign> comes first and the <foreign
   lang="la">quaesitum</foreign> second, the <foreign lang="la">datum</foreign>
   being in this proposition the greater <em>side</em> and in the next the
   greater <em>angle</em>. Thus the enunciations are (<a href="/elem.1.18">I.
   18</a>) <foreign lang="greek">παντὸς τριγώνου ἡ μείζων πλευρὰ τὴν μείζονα
   γωνίαν ὑποτείνει</foreign> and (<a href="/elem.1.19">I. 19</a>) <foreign
   lang="greek">παντὸς τριγώνου ὑπὸ τὴν μείζονα γωνίαν ἡ μείζων πλευρὰ
   ὑποτείνει</foreign>. In order to keep the proper order in English we must
   use the passive of the verb in <a href="/elem.1.19">I. 19</a>. Aristotle
   quotes the result of <a href="/elem.1.19">I. 19</a>, using the exact
   wording, <foreign lang="greek">ὑπὸ γὰρ τὴν μείζω γωνίαν ὑποτείνει</foreign>
   (<xref n="Aristot. Meteo.. 376a.12" from="ROOT"
   to="DITTO"><title>Meteorologica</title> III. 5, 376 a 12</xref>).