Efficient and correct sampling of bond angle distributions for molecular models with fixed bond length constraints present considerable challenges in Monte Carlo simulations. Most of the algorithms developed to date address the issue by focusing on internal coordinate generation and thus are difficult to implement in an existing Monte Carlo code. In this presentation, we demonstrate an exceedingly simple and efficient MC algorithm that correctly generates bond angle distributions at a branch point. This is achieved by perturbing Cartesian coordinates of a randomly chosen atom of the branch point. The algorithm is designed to preserve the fixed bond length constraint at each step. Results obtained from the algorithm for bond angle distributions for branch points containing three and four branches have been validated against those produced by a brute force Boltzmann rejection scheme at 298 and 1000 K. Extension of the method in combination with ring closure will also be discussed for efficient sampling of cyclic molecules with constrained bond lengths such as an all-atom model of benzene.