  We investigate two one-sided orthogonalities of matrices, the first of which
is left (right) $*$-orthogonality for rectangular matrices and the other is
left (right) core-orthogonality of index $1$ matrices. We obtain some basic
results for these matrices, their canonical forms, and characterizations. Also,
relations between left (right) orthogonal matrices and parallel sums are
investigated. Finally under these one-sided orthogonalities we explore the
conditions of additivity of the Moore-Penrose inverse and the core inverse.
