1961 AMC 12 Problem 27

An equiangular $n$-gon has angle $180^\circ-\frac{360^\circ}{n}$

Use $kx<180^\circ$ and $k\ge2$ to constrain the smaller angle $x$

Every polygon angle is at least $60^\circ,$ and every convex polygon angle is less than $180^\circ.$ Since $k\ge2$ and $kx<180^\circ,$ we need $x<90^\circ.$ The only possible equiangular polygon angle in $[60^\circ,90^\circ)$ is the triangle angle $x=60^\circ.$ Then $k=2$ gives $kx=120^\circ,$ the angle of a regular hexagon; larger $k$ is impossible. Hence there is one pair.

Thus, the correct answer is D.