  Following Beck's work on toral translations relative to straight boxes in
$\mathbb{T}^n$, we prove a weaker upper bound and the same lower bound for
ergodic discrepancies of toral translations relative to a triangle in
$\mathbb{T}^2$. Specifically, given a positive increasing function
$\varphi(n)$, we show that for a full measure set of translation vectors
$\alpha\in \mathbb{T}^2$, if the series $\sum_{N=1}^{\infty} \frac{1}{\varphi
(N)} $ converges, then the maximal discrepancy of toral translations relative
to the triangles of a given slope $\tau$ is bounded from above by
$Const(\alpha,\tau) (\log N)^2 \varphi^2(\log \log N)$, and there would be
infinitely many $N$'s such that the maximal discrepancy is greater than $(\log
N)^2\varphi(\log \log N)$ if the series $\sum_{N=1}^{\infty} \frac{1}{\varphi
(N)} $ diverges. An important difference between our result and that of Beck'
is an additional factor $\varphi(\log \log N)$, which is necessary in our proof
for controlling the new small divisors created by the hypotenuse.
