We study the properties and thermodynamic stability of the plane symmetry black hole from the viewpoint of geometry. We find that the Weinhold curvature gives the first-order phase transition at N=1, where N is a parameter of the plane symmetry black hole while the Ruppeiner one shows first-order phase transition points for arbitrary N≠1. Considering the Legendre invariant proposed by Quevedo et al., we obtain a unified geometry metric, which contains the information of the second-order phase transition. So, the first-order and second-order phase transitions can be both reproduced from the geometry curvatures. The geometry is also found to be curved, and the scalar curvature goes to negative infinity at the Davie phase transition points beyond semiclassical approximation.