This Demonstration shows three spherical triangles. Triangles and have a common base and the same area, so they are equivalent by dissection (as shown in Dissection of Two Spherical Triangles with a Common Base). Triangles and also have a common base and the same area, so they are also equivalent by dissection. So triangles and are equivalent by dissection.

Therefore, spherical triangles with the same area are equivalent by dissection.

[1] P. Gerwien, "Zerschneidung jeder beliebigen Anzahl von gleichen geradlinigen Figuren in dieselben Stücke," Journal für die reine und angewandte Mathematik (Crelle's Journal), 10, 1833 pp. 228–234, and Taf. 3. gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002139170.