We develop, and analyze two distributed event-triggered linear iterative algorithms that enable the components of a distributed system, each with some initial value, to reach approximate average consensus on their initial values, after executing a finite number of iterations. Each proposed algorithm provides a criterion that allows the nodes to determine, in a distributed manner, when to terminate because approximate average consensus has been reached, i.e., all nodes have obtained a value that is within a small distance from the average of their initial values. We focus on a distributed system whose underlying topology is captured by an undirected (symmetric) graph, and develop linear iterative strategies with time-varying weights, chosen based on the subset of edges that separate nodes with significantly different values and are considered active at each iteration. In our simulations, we illustrate the proposed algorithms and compare the number of iterations and transmitted values required by the proposed protocols against a previously proposed stopping protocol for approximate average consensus. (C) 2016 Elsevier Ltd. All rights reserved.