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In this work, we present forecasts for the joint analysis of <jats:italic>Euclid<\/jats:italic> and CMB data on the cosmological parameters of the standard cosmological model and some of its extensions. This work expands and complements the recently published forecasts based on <jats:italic>Euclid<\/jats:italic>-specific probes, namely galaxy clustering, weak lensing, and their cross-correlation. With some assumptions on the specifications of current and future CMB experiments, the predicted constraints are obtained from both a standard Fisher formalism and a posterior-fitting approach based on actual CMB data. Compared to a <jats:italic>Euclid<\/jats:italic>-only analysis, the addition of CMB data leads to a substantial impact on constraints for all cosmological parameters of the standard \u039b-cold-dark-matter model, with improvements reaching up to a factor of ten. For the parameters of extended models, which include a redshift-dependent dark energy equation of state, non-zero curvature, and a phenomenological modification of gravity, improvements can be of the order of two to three, reaching higher than ten in some cases. 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