The exterior angle theorem states that an exterior angle of a triangle in Euclidean geometry equals the sum of the two nonadjacent interior angles. Here the exterior angle is formed by a side and the extension of an adjacent side. This is Proposition I.32 of Euclid's Elements (Heath 1956).
To prove the equality, let 
, 
, and 
be the interior angles
of the triangle, measured in radians,
and let 
be the exterior angle adjacent to 
. The triangle's interior
angles satisfy 
. Since 
and 
are supplementary
angles,
 |
The name exterior angle theorem is also used for the weaker statement that an exterior angle of a triangle
in Euclidean geometry is greater than either
nonadjacent interior angle individually. In the
notation above, 
and 
.
This is Proposition I.16 of Euclid's Elements,
which is proved without the parallel postulate.
The equality in Proposition I.32 uses that postulate (Heath 1956).
