If two commensurable magnitudes be added together, the whole will also be commensurable with each of them; and if the whole be commensurable with one of them, the original magnitudes will also be commensurable.
Proof
a,b have a common measure d, so a+b=(m+n)d shares d.
Converse: if a+b and a share a measure, then so does b by
Common Notion 3.