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In this paper we extend the concept of coherent pairs of measures from the real line to Jordan arcs and curves. We present a characterization of pairs of coherent measures on the unit circle: it is established that if (mu0,mu1) is a coherent pair of measures on the unit circle, then mu0 is a semi-classical measure. Moreover, we obtain that the linear functional associated with mu1 is a specific rational transformation of the linear functional corresponding to mu0. Some examples are given.