The generalized triangle groups are two generator groups with a presentation \(a^ n=b^ p=R^ m(a,b)=1\) where \(2\leq n\leq p\), \(m\geq 2\) and R(a,b) is a cyclically reduced word involving both a and b. For the usual triangle groups where \(R(a,b)=ab\) it is well-known for which triples of integers n, p, m the group is infinite. The authors show that these generalized triangle groups are infinite when \(m\geq 3\) and \(\frac{1}{m}+\frac{1}{n}+\frac{1}{p}\leq 1\) and also for most cases when \(m=2\). The method of proof is to choose elements A and B(z) in \(SL_ 2({\mathbb{C}})\) whose orders are n and p in \(PSL_ 2({\mathbb{C}})\) and then choose \(z\in {\mathbb{C}}\) such that tr R(a,b) is 2 cos(k\(\pi\) /m), \((k,m)=1\), thereby obtaining a representation of the group in \(PSL_ 2({\mathbb{C}})\). A more judicious choice of z shows that the image cannot be one of the (well-known) two generator finite groups in \(PSL_ 2({\mathbb{C}})\).