A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

12 Spherical Geometry.
15. The sides of either af two polar triangles are the supplements of the angles of the other.
DEi.-Produce the arcs A'B', A'C' if necessary to meet BC
in E and F. Now, since C is the pole of 'B', the arc EC is
90~. In like manner, the arc BF is 90~. Ience EC + BF or
BC+ EFis 180~; but EF is (Art. 8, Cor. 1) the measure
of the spherical angle B'A'C'. Hence the side BC is the supplement of the angle B'A' C', and similarly for the other sides
and angles.
Scholium.-On account of the property proved in this proposition, polar triangles are also called supplemental triangles.
Cor.-If we denote the angles of the triangle ABC by
A, B, C; their opposite sides by a, b, c; and the corresponding
elements in the polar triangle by the same letters accented, we
have
a'= 180 - A, '= 180 - a. (4)
b' = 1800 - B, B' = 180~ - b. (5)
' =180~- C, C' = 180-. (6)
16. In every spherical triangle-1~. The sum of two angles is
less than the third increased by 180~. 2~. The sum of the three
ngles is greater than two, and less than six right angles.
DEM.-1~. From the equations (4)-(6) we get
(a + b' - c') = 180 + C- (A + B);
but a' + b' is greater than e', therefore 180~ + C is greater than
A +B.
2~. From the same equations, we have
A + B + C= 540~ - (a' + b' + c').
Hence A + B + C is less than 540~; that is, six right angles.
Again (from ~ 12), a' + b' + c' is less than 360~. Hence
A + B + C is greater than 180~; that is, two right angles.