  We prove that if the Hausdorff dimension of a compact subset of ${\mathbb
R}^d$ is greater than $\frac{d+1}{2}$, then the set of angles determined by
triples of points from this set has positive Lebesgue measure. Sobolev bounds
for bi-linear analogs of generalized Radon transforms and the method of
stationary phase play a key role. These results complement those of V. Harangi,
T. Keleti, G. Kiss, P. Maga, P. Mattila and B. Stenner in (\cite{HKKMMS10}). We
also obtain new upper bounds for the number of times an angle can occur among
$N$ points in ${\mathbb R}^d$, $d \ge 4$, motivated by the results of Apfelbaum
and Sharir (\cite{AS05}) and Pach and Sharir (\cite{PS92}). We then use this
result to establish sharpness results in the continuous setting. Another
sharpness result relies on the distribution of lattice points on large spheres
in higher dimensions.
