This study explores a gas measurement method based on Fabry-Perot (F-P) angle-dependent correlated spectroscopy, which achieves highly sensitive and selective gas measurements by adjusting the angle to match the F-P interference peak with the gas absorption peak. Methane (CH4) is the chosen target gas, and an F-P etalon is designed with parameters matching the CH4 absorption peak. An angle-scanning measurement system is established to enable correlated spectroscopic detection of CH4 gas. Angle-scanning measurements reveal distinct absorption signals at the angle where the F-P interference peak aligns with the CH4 absorption peak. Gas measurements of standard samples demonstrate a linear relationship between the apparent absorbance at the on/off positions and CH4 concentration, allowing for accurate CH4 concentration measurements. The study further investigates the detection limit of the experimental system, achieving a 3σ detection limit of 720 ppm under the on/off measurement mode. A conical incidence model is developed to analyze the impact of beam divergence angles on the transmittance of the F-P cavity. Simulations are conducted to assess absorption signals in the presence of extreme cross-interference, demonstrating the method's robust resistance to cross-interference. The F-P correlated spectroscopy method described in this paper, as a non-dispersive spectroscopic measurement technique, holds promise for designing high-sensitivity gas sensors and imaging applications.