Commensurability of Sum of Commensurable Magnitudes

Proof
Let $AB$ and $BC$ be commensurable magnitudes which are added together to make $AC$.

Since $AB$ and $BC$ are commensurable, then some magnitude $D$ will measure them both.

Since $D$ measures both $AB$ and $BC$, $D$ also measures $AC$.

That is, $D$ measures $AB$, $BC$ and $AC$.

Therefore from, $AC$ is commensurable with each of the magnitudes $AB$ and $BC$.

Let $AC$ be commensurable with $AB$.

Since $AC$ and $AB$ are commensurable, then some magnitude $D$ will measure them both.

Since $D$ measures both $AC$ and $AB$, $D$ also measures the remainder $BC$.

That is, $D$ measures $AB$, $BC$ and $AC$.

Therefore from, $AB$ and $BC$ are commensurable.