Napier's Rules for Right Angled Spherical Triangles

Theorem
Napier's Rules for Right Angled Spherical Triangles are the special cases of the Spherical Law of Cosines for a spherical triangle one of whose angles or sides is a right angle.
 * NapiersRules.png

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.

Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.

Let either angle $\angle C$ or side $c$ be a right angle.

Let the remaining parts of $\triangle ABC$ be arranged in a circle as above:
 * for $\angle C$ a right angle, the interior
 * for $c$ a right angle, the exterior

where the symbol $\Box$ denotes a right angle.

Let one of the parts of this circle be called a middle part.

Let the two neighboring parts of the middle part be called adjacent parts.

Let the remaining two parts be called opposite parts.

Also known as
This suite of rules is also known as Napier's rules of circular parts.