  In this paper, we study a close relationship between relative cluster tilting
theory in extriangulated categories and tau-tilting theory in module
categories. Our main results show that relative rigid objects are in bijection
with $\tau$-rigid pairs, and also relative maximal rigid objects with support
tau-tilting pairs under some assumptions. These results generalize their work
by Adachi-Iyama-Reiten, Yang-Zhu and Fu-Geng-Liu. Finally, we introduce
mutation of relative maximal rigid objects and show that any basic relative
almost maximal rigid object has exactly two non-isomorphic indecomposable
complements.
