In the sphere above, the shaded area defined by the points A, B, C clearly makes a triangle. My question is, can the complement of this area, that is everything on the sphere that is white, also be considered a triangle? It is continuous and contains three edges.
You can do more than that. Triangles can be defined by reversing one, two or all three sides of what we ordinarily consider triangles, provided we use the proper supplementary or reflex angles to define the "interior" angles corresponding to the side orientations.
We illustrate with a triangle $ABC$ with $A$ at the projected face center of an inscribed cube, $B$ at an adjacent vertex and $C$ at an edge center adjacent to the face center $A$ and vertex $B$. From the geometric construction the angles at $A,B,C$ respectively measure $45°,60°,90°$ ad the respective opposing sides measure $35°,45°,55°$; the sides are rounded to the nearest degree for brevity.
Now say we reverse side $a$, going the long way around the sphere from $B$ to $C$. Now $a$ will gave the reflex measure $325°$, angles $B$ and $C$ becone their supplentary values $120°$ and $90°$, and angle $A$ measured on the same side of the perimeter as angles $B$ and $C$ takes on the reflex value $315°$. Sides $b$ and $c$ remain $45°$ and $55°$. Strange as these numbers may seem, if you plug them into spherical trigonometric laws such as the Law of Sines and the Laws of Cosines, they fit perfectly well apart from the rounding mentioned earlier.
One thing that does change is the angle sum: it was $195°$ for the original triangle but now is $525°$ for the interior angles as defined in this paragraph. Reversing side $a$ leads to a different, in this case larger, portion of the sphere being enclosed with the supplementary angles at $B$ and $C$ and the reflex angle at $A$.
We may similarly render:
$a,b$ reversed: $A,B,C=325°,315°,55°;a,b,c=135°,120°,270°$
$a,b,c$ reversed: $A,B,C=325°,315°,305°;a,b,c=315°,240°,270°$
We may do similar things in Euclidean geometry, exchanging the triangle for a figure containing open rays along the sides that were reversed or turned inside-out. For example, in what we usually call the $3-4-5$ right triangle the angles measure $37°,53°,90°$ (rounded where necessary). Now draw the hypotenuse the "wrong" way, forming rays that lie along the line between the vertices but going to infinity instead of between the vertices. Taking the supplementary angles on the reversed hypotenuse as interior angles, they now measure $143°,127°$. The right angle, measured on the same side of the boundary, now measures $270°$. The Laws of Sines and Cosines are now consistent with the sides measuring $3,4,-5$ instead of $3,4,+5$; the reversed orientation corresponds to the length effectively becoming negative. The Pythagorean Theorem, conveniently relating only even powers of the sides, still holds.