Independent random variables $(X_i)$ have the same law, then $(X_i,\sum X_i ) $ have the same law for any $i$

Question

Let $(X_i)$ be sequence of real independent r.v.'s and having the same law. If we let $X=\sum X_i$, how can one show that $(X_i, X)$ have the same law for any $i$. In this case $(X_i, X)$ is a random vector. (I think we can also say if $(X_i, X )$ have the same law then every $X_i$ have the same law. But no idea how to prove this too). Since $X$ and $X_i$ are not independent we can't do the usual trick to seperate the laws.