the greater side of a triangle subtends the greater angle

I.18

In any triangle the greater side subtends the greater angle.

—Euclid

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

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.

## 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>).