This paper investigates the state consensus of linear multi-agent systems in a graph where each agent is equipped with two novel event-triggering mechanisms. Each agent utilizes them to avoid continuous information transmissions with its neighbors and to reduce the frequencies of controller updates, respectively. One of the event-triggering mechanisms defines a threshold of state errors by a constant plus a state-dependent variable. The other event-triggering mechanism introduces a period of rest time after each event. For each agent, both event-triggering mechanisms are fully distributed and are independent of any global information. The authors utilize a co-design approach to deal with the interplay between control gains and parameters in event-triggering mechanisms. With appropriate control gains in control laws and parameters in event-triggering conditions, subsystems employing discrete-time signals from neighbors and discrete-time signals from their controllers achieve the state consensus. Simulations are performed to illustrate the effectiveness of the proposed event-triggering mechanisms.