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

14 Spherical Geometry.
small circle, passing through A, B, C, and P' the pole of the
circle through A4'B'C'; then evidently P' is diametrically opposite to P, and the pairs of triangles PAB, P'B'A'; PB C, P' C'B';
PCA, P'A'C', being antipodal and isosceles, are superposable.
Hence the area of AB C is equal to the area of A'B' C'.
18. Two spherical triangles on the same sphere have all their
corresponding elements equal-l1~. When two sides and the contained angle of one are respectively equal to two sides and the contained angle of the other. 2~. Whien the side and the adjacent
angles of one are equal to a side and the adjacent angles of the
other. 3~. When the three sides of one are equal to the three
sides of the other. 4~. When the three angles of one are equal
to the three angles of the other.
'i
Fig. 8.
Cases 1~, 2~, 3~ correspond to Euc. Book I., Props. Iv., vIII.,
xxvI. Case 4~ has no analogue in Plane Geometry. It will be
sufficient to prove 1~ and 3~, as 2~ and 4~ are inferred from
them by the properties of the supplemental triangle.
DEM. 1~.-A = A', AB = A'B'; AO = A'C'. If these elements are arranged in the same order, the demonstration follows
by superposition, as in Plane Geometry. If they are disposed
in an inverse order, such as A'B'C', B C", we can superpose
either of them on the antipodal triangle of the other.