Solution

Suppose the right endpoint of the rod is at $x=x'=0$, $y=y'=0$ at time $t=t'=0$ according to Observer $O$ on the floor and Observer $O'$ moving to the right with the speed $\beta_1c$ relative to the floor. According to Observer $O'$, who makes measurement of the rod at time $t'=0$, the left endpoint of the rod is at $x'=-L/\gamma$ (due to length contraction) and $y'$ to be determined, where $L$ is the length of the rod according to Observer $O$ and $\gamma=1/\sqrt{1-\beta_1^2}$. Observer $O$ records this left-endpoint event at time

$$t=\gamma\left(t'+\frac{\beta_1x'}{c}\right)=-\frac{\beta_1L}{c}$$

and hence at $y=\beta\beta_1L$ (and $x=-L$), because the rod is moving downward with the speed $\beta c$. Since $y'=y$, Observer $O'$ witnesses the rod making an angle

$$\arctan\frac{y'}{|x'|}=\arctan\frac{\beta\beta_1}{\sqrt{1-\beta_1^2}}$$

with the floor.