  We prove that if $E \subset {\mathbb F}_q^2$, $q \equiv 3 \mod 4$, has size
greater than $Cq^{7/4}$, then $E$ determines a positive proportion of all
congruence classes of triangles in ${\mathbb F}_q^2$.
  The approach in this paper is based on the approach to the Erd\H os distance
problem in the plane due to Elekes and Sharir, followed by an incidence bound
for points and lines in ${\mathbb F}_q^3$. We also establish a weak lower bound
for a related problem in the sense that any subset $E$ of ${\mathbb F}_q^2$ of
size less than $cq^{4/3}$ definitely does not contain a positive proportion of
{\bf translation} classes of triangles in the plane. This result is a special
case of a result established for $n$-simplices in ${\mathbb F}_q^d$. Finally, a
necessary and sufficient condition on the lengths of a triangle for it to exist
in $\mathbb{F}^2$ for any field $\mathbb F$ of characteristic not equal to 2 is
established as a special case of a result for $d$-simplices in ${\mathbb F}^d$.
