52A40

The sines of -dimensional vertex angles of an -simplex is defined and the law of sines for -dimensional vertex angles of an -simplex is established. Using the generalized sine law for -simplex, we obtain some inequalities for the sines of -dimensional vertex angles of an -simplex. Besides, we obtain inequalities for volumes of -simplices. As corollaries, the generalizations to several dimensions of the Neuberg-Pedoe inequality and P. Chiakuei inequality, and an inequality for pedal simplex are given.;