It is known that $\mathrm{x}+\mathrm{y}=10,$then $\mathrm{x}+\mathrm{y}+\mathrm{z}=10+\mathrm{z}$.

The Euclid's axiom that illustrates this statement is:

Given, $\mathrm{x}+\mathrm{y}=10,\mathrm{ }$then $\mathrm{x}+\mathrm{y}+\mathrm{z}=10+\mathrm{z}$.
Here $\mathit{z }$has been added to two quantities that are equal. 
So, by Euclid's second axiom, which states that, 
If equals are added to equals then wholes are equal, 
$\Rightarrow \mathrm{x}+\mathrm{y}+\mathrm{z}=10+\mathrm{z}$ 

Hence, $\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{+}\mathbf{z}\mathbf{=}\mathbf{10}\mathbf{+}\mathbf{z}$ is true by the second axiom of Euclid which is, “any straight line segment can be extended indefinitely in a straight line”.