  We introduce a new generalization of $\theta$-congruent numbers by defining
the notion of rational $\theta$-parallelogram envelope for a positive integer
$n$, where $\theta \in (0, \pi)$ is an angle with rational cosine. Then, we
study more closely some problems related to the rational $\theta$-parallelogram
envelopes, using the arithmetic of algebraic curves. Our results generalize the
recent work of T.~Ochiai, where only the case $\theta=\pi/2$ was considered.
Moreover, we answer the open questions in his paper and their generalizations
for any Pythagorean angle.
