Independence of two $\chi^2$ distribution

Question

Suppose we are given that $X_1$ and $X_2$ are two non-negative random variable such that $$X_1+X_2\sim\chi^2_{(2)}$$. Also we are given that $X_1\sim\chi^2_{(1)}$. Can we say the following

1. $X_2$ is independent of $X_1$?
2. $X_2\sim\chi^2_{(1)}$? If not, then please provide a counterexample.

This is just a fun question question. I'm asking it just for curiosity. Any kind of help appreciated.