We study the elastic response of concentrated suspensions of rigid wire frame particles to a step strain. These particles are constructed from infinitely thin, rigid rods of length L. We specifically compare straight rod-like particles to bent and branched wire frames. In dense suspensions, the wire frames are frozen in a disordered state by the topological entanglements between their arms. We present a simple, geometric method to find the scaling of the elastic stress with concentration in these glassy systems. We apply this method to a simple 2D model system where a test particle is placed on a plane and constrained by a random distribution of points with number density ν. Two striking differences between wire frame and rod suspensions are found: (1) The linear elasticity per particle for wire frames is very large, scaling like ν2L4, whereas for rods, it is much smaller and independent of concentration. (2) Rods always shear thin but wire frames shear harden for concentrations less than ∼K/kBTL4, where K is the bending modulus of the particles. The deformation of wire frames is found to be important even for small strains, with the proportion of deformed particles at a particular strain, γ, being given by (νL2)2γ2. Our results agree well with simple numerical calculations for the 2D system.