  We introduce the higher version of the notion of Adachi-Iyama-Reiten's
support $\tau$-tilting pairs, which is a generalization of maximal
$\tau_n$-rigid pairs in the sense of Jacobsen-J{\o}rgensen. Let $\mathcal C$ be
an $(n+2)$-angulated category with an $n$-suspension functor $\Sigma^n$ and an
Opperman-Thomas cluster tilting object. We show that relative $n$-rigid objects
in $\mathcal C$ are in bijection with $\tau_n$-rigid pairs in the $n$-abelian
category $\mathcal C/{\rm add}\Sigma^n T$, and relative maximal $n$-rigid
objects in $\mathcal C$ are in bijection with support $\tau_n$-tilting pairs.
We also show that relative $n$-self-perpendicular objects are in bijection with
maximal $\tau_n$-rigid pairs. These results generalise the work for $\mathcal
C$ being $2n$-Calabi-Yau by Jacobsen-J{\o}rgensen and the work for $n=1$ by
Yang-Zhu.
