Silicon nitride (Si₃N₄) optical ring resonators provide exceptional opportunities for low-loss integrated optics. Here we study the transmission through a multimode waveguide coupled to a Si₃N₄ ring resonator. By coupling single-mode fibers to both input and output ports of the waveguide we selectively excite and probe combinations of modes in the waveguide. Strong asymmetric Fano resonances are observed and the degree of asymmetry can be tuned through the positions of the input and output fibers. The Fano resonance results from the interference between modes of the waveguide and light that couples resonantly to the ring resonator. We develop a theoretical model based on the coupled mode theory to describe the experimental results. The large extension of the optical modes out of the Si₃N₄ core makes this system promising for sensing applications.