Do alternating and augmented paths for a matching need to cover all the edges in the matching?

Definition for alternating paths and augmented paths of a matching in a graph is defined as follows:

Given a matching M, an alternating path is a path in which the edges belong alternatively to the matching and not to the matching. an augmenting path is an alternating path that starts from and ends on free (unmatched) vertices.

I think both concepts are not required to include all the edges of the matching. So from any alternating path, we can find a part of it which can be an augmenting path. But this would contradict a theorem that a matching is maximum if and only if there is no augmented path for it. So I wonder how I should understand the two concepts?

Thanks!

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