The question asks to identify the incorrect geometry among the listed molecules based on their electron geometry.
1. Step 1: Understand the molecular shapes.
- BrF$_5$: The molecule has 5 fluorine atoms attached to a central bromine atom, with one lone pair of electrons on the bromine. According to the VSEPR theory, BrF$_5$ adopts an octahedral geometry, not trigonal bipyramidal. Therefore, the statement for BrF$_5$ is incorrect.
- SF$_4$: This molecule adopts a see-saw geometry due to 4 bonding pairs and one lone pair of electrons on sulfur, which is correct.
- NH$_3$: Ammonia adopts a pyramidal shape due to 3 bonding pairs and one lone pair of electrons on nitrogen, which is correct.
- XeF$_4$: Xenon tetrafluoride has a square planar geometry because it has 4 bonding pairs and 2 lone pairs of electrons, making the geometry planar, which is correct.
Thus, the incorrect geometry is for BrF$_5$, and the correct geometry is octahedral.