Surface family pair with Bertrand pair as mutual geodesic curves in Euclidean 3-space $ \mathbb{E}^{3} $

The main interest of this work is to construct surface family pair with the symmetry of Bertrand pair in Euclidean 3-space $ \mathbb{E}^{3} $. Then, by employing the Serret-Frenet frame, we conclude the sufficient and necessary conditions of surface family pair interpolating Bertrand pair as mutual geodesic curves. Moreover, the conclusion to ruled surface family pair is also obtained. Meanwhile, this work is demonstrated through several examples.