This study investigates the consensus control problem of linear multi-agent systems (MASs) with network-induced delays (NIDs). A state feedback controller based on an event-triggered transmission scheme is proposed, where the novel adaptive event-triggering mechanism (ETM) is used to lower the frequency of data transmission. In contrast to previous works, sufficient conditions related to the consensus for the resulting closed-loop control systems are derived using a discontinuous looped Lyapunov functional (DLLF) and the corresponding reciprocally convex protocol. It is demonstrated that the consensus errors of resulting closed-loop control systems are asymptotically stable and the Zeno behavior is excluded. Finally, a numerical example of an unmanned aerial vehicle (UAV) model is presented to demonstrate the effectiveness of our proposed theoretical results.