A quickest change detection (QCD) algorithm is proposed to address the problem of detecting and identifying line outages in a power system. The statistics of electricity demand are assumed to be known and propagated through a linearized model of the equations describing the power flow balance at each node of the network. The proposed algorithm incorporates the transient dynamics of the power system following a line outage and is applied to the measurements of voltage phase angles, which are collected using phasor measurement units. This adaptive algorithm is developed by treating the QCD problem as a dynamic composite hypothesis testing problem. It is shown to have superior performance compared with other line outage detection algorithms previously proposed in the literature. Case studies demonstrating this gain in performance are illustrated through the IEEE 118-bus test system.