$(a,b) R (c,d)$ iff $a+d = b+c$.
So,
$(c,d) R (a,b)$ iff $c+b = d+a$.
From property of addition, $d+a = a + d, c+b = b+c$.
So, $(a,b) R (c,d)$ iff $(c,d) R (a,b)$. Hence, $R$ is symmetric.
$a+b = b+a$, so $(a,b) R (a,b)$. Hence, $R$ is reflexive.
If, $(a,b) R (c,d)$ and $(c,d) R (e,f)$, we have $a+d = b+c$ and $c+f = d+e$. So, $a + d + c + f = b + c + d + e \implies a + f = b + e \implies (a,b) R (e,f).$
Hence, $R$ is transitive.
Since $R$ is reflexive, transitive and symmetric, it is an equivalence relation.