  In this note, we extend the main results of our paper on quasilinearization
and curvature of Aleksandrov spaces of curvature $\leq0$ to curvature bounds
other than $0$. For non-zero $K$, we employ the previously introduced notion of
the $K$-quadrilateral cosine, which is the cosine under parallel transport in
model $K$-space, and which is denoted by $\operatorname{cosq}_{K}$. Our
principal result states that a geodesically connected metric space (of diameter
not greater than $\pi/\left(2\sqrt{K}\right) $ if $K>0$) is an $\Re_{K}$ domain
(otherwise known as a $\operatorname{CAT}\left(K\right) $ space) if and only if
always $\operatorname{cosq}_{K}\leq1$ or always $\operatorname{cosq}_{K}$
$\geq-1$. (We prove that in such spaces always $\operatorname{cosq}_{K}\leq1$
is equivalent to always $\operatorname{cosq}% _{K}$ $\geq-1$). As a corollary,
we give necessary and sufficient conditions for a Cauchy complete semimetric
space to be a complete $\Re_{K}$ domain. We show that in our theorem the
diameter hypothesis for positive $K$ is sharp and we prove an extremal theorem
when $|\operatorname{cosq}_{K}|$ attains an upper bound of $1$. We derive from
our main theorem and our previous result for $K=0$ a complete solution of
Gromov's curvature problem in the context of Aleksandrov spaces of curvature
bounded above. Then we establish the $K$-Euler's inequality and the extremal
theorem for equality in the $K$-Euler's inequality in an $\Re_{K}$ domain.
