  Let ${\mathbb K}={\mathbb Q}(\sqrt{m})$ be a real quadratic number field,
where $m>1$ is a squarefree integer. Suppose that $0 < \theta< \pi $ has
rational cosine, say $\cos (\theta)=s/r$ with $0< |s|<r$ and $\gcd(r,s)=1$. A
positive integer $n$ is called a $(\mathbb K,\theta)$-congruent number if there
is a triangle, called the $(\mathbb K,\theta, n)$-triangles, with sides in
$\mathbb K$ having $\theta$ as an angle and $n\alpha_\theta$ as area, where
${\alpha_\theta}=\sqrt{r^2-s^2}$. Consider the $(\mathbb K,\theta)$-congruent
number elliptic curve $E_{n,\theta}: y^2=x(x+(r+s)n)(x-(r-s)n)$ defined over
$\mathbb K$. Denote the squarefree part of positive integer $t$ by ${\rm
sqf}(t)$. In this work, it is proved that if $m\neq {\rm sqf}(2r(r-s))$ and
$mn\neq 2, 3, 6$, then $n$ is a $(\mathbb K,\theta)$-congruent number if and
only if the Mordell-Weil group $E_{n,\theta}(\mathbb K)$ has positive rank, and
all of the $(\mathbb K,\theta, n)$-triangles are classified in four types.
