The following selections from Regiomontanus: On Triangles (email bradley at adelphi dot edu for page scans):
Book III, theorems 2, 3 [pages 144-145]
Book III, theorem 16 [pages 158-159]
Book IV, theorem 4 [pages 206-207]
Book IV, theorem 15 : Look at the theorem statement only (not the proof). [pages 218-221] A version of the "theorem that dispenses" or "rule of four quantities"
Book IV, theorems 19 [pages 230-233]
Book V, theorem 1 [pages 268-271]
Book V, theorem 2 : Look at the theorem statement only (not the proof). [pages 270-275, including proof. pages 270-271 for Results without proof]. Equivalent to spherical law of cosines for sides.
Optional readings, time and interest permitting:
Book IV, theorem 16 [pages 222-225] spherical law of sines for right triangles
Book IV, theorem 17 [pages 225-227] spherical law of sines for non-right triangles
Book IV, theorem 18 [pages 227-231] another one of Napier's rules