Due to lack of the third dimension in 3D bulk materials, the crack tip in graphene locates on several atoms implying that its fracture behavior can be closely associated with its lattice structure, i.e., the bond length and angle. As the bond length reflects the discrete nature of the atomic structure, theoretical discussion is focused on the concomitant size effect at the nanoscale with few or no reports about the influence of the bond angle. Through the comparisons between theoretical calculations and experimental data, here it is first demonstrated that the bond angle is essential for understanding the fracture behavior in graphene, serving as an intrinsic notch reducing the stress singularity near the crack tip (the intrinsic notch effect), leading to the breakdown of the Griffith criterion in graphene. The work provides a framework for the studying of the brittle fracture in 2D materials, which gives rise to the more reliable device design based on 2D materials. More importantly, the significance of the intrinsic notch effect is profound and far-reaching, paving the way to a more comprehensive and deep understanding of the mechanical properties in nano as well as nanostructured materials.