Page:Spherical Trigonometry (1914).djvu/32

is, the angle subtended by $B'C'$ at the centre of the sphere is the supplement of the angle $A$. This we may express for shortness thus; $B'C'$ is the supplement of $A$. Similarly it may be shewn that $C'A'$ is the supplement of $B$, and $A'B'$ the supplement of $C$.

And since $ABC$ is the polar triangle of $A'B'C'$, it follows that $BC$, $CA$, $AB$ are respectively the supplements of $A'$, $B'$, $C'$; that is, $A'$, $B'$, $C'$ are respectively the supplements of $BC$, $CA$, $AB$.

From these properties a primitive triangle and its polar triangle are sometimes called supplemental triangles.

Thus, if $A$, $B$, $C$, $a$, $b$, $c$ denote respectively the angles and the sides of a spherical triangle, all expressed in circular measure, and $A'$, $B'$, $C'$, $a'$, $b'$, $c'$ those of the polar triangle, we have $${\begin{aligned}A'&=\pi -a,&B'&=\pi -b,&C'&=\pi -c,\\a'&=\pi -A,&b'&=\pi -B,&c'&=\pi -C.\end{aligned}}$$

28. Duality of theorems relating to the spherical triangle.

The preceding result is of great importance; for if any general theorem be demonstrated with respect to the sides and the angles of any spherical triangle it holds of course for the polar triangle also. Thus any such theorem will remain true when the angles are changed into the supplements of the corresponding sides and the sides into the supplements of the corresponding angles. We shall see several examples of this principle in the next Chapter.

29. Any two sides of a spherical triangle are together greater than the third side. (See the figure of Art. 18.)

For any two of the three plane angles which form the solid angle at $O$ are together greater than the third (Euclid, XI. 20). Therefore any two of the arcs $AB$, $BC$, $CA$, are together greater than the third.

From this proposition it is obvious that any side of a spherical triangle is greater than the difference of the other two.