While piecewise geometric Brownian motion is a stochastic process that can effectively incorporate discrete dividends into stock prices without losing consistency, the process results in the lack of closed-form solutions for option prices. We aim to resolve this by providing analytical approximation formulas for American call option prices under this process. Our approximations include lower and upper bounds of the option prices. The additional numerical analysis indicates that the lower bound method is very efficient and it is possible to reduce the pricing error by considering additional terms.