Bond angles around a tetravalent atom

Relationships among the six bond angles about a central tetravalent atom depend on symmetry, ranging from the most symmetrical Td point group to the least symmetrical C1 point group having only the identity element. Exact relationships are derived here in two ways: (1) a purely algebraic treatment of the general mathematical conditions among the bond angles, followed by factorizations that arise from various symmetry constraints and (2) a reverse approach based on geometric analysis, starting with the most symmetrical Td case and relaxing constraints stepwise to lower point groups. The mathematical formulas show systematically how the degrees of freedom among the bond angles increase from zero to a maximum of five as the symmetry is relaxed from the Td symmetry.