This letter characterizes the contact angle distribution based on the condition that the relay low earth orbit (LEO) satellite is in the communication range of both the ground transmitter and the ground receiver. As one of the core distributions in stochastic geometry-based routing analysis, the analytical expression of the Cumulative Distribution Function (CDF) of the conditional contact angle is derived. Furthermore, the conditional contact angle is applied to analyze the inaccessibility of common satellites between the ground transmitter and receiver. Finally, with the help of the conditional contact angle, coverage probability and achievable data rate in LEO satellite-relayed transmission are studied.