jmc

Solving $\frac{1}{3}\pi (12\text{ cm}{)}^2(h)=432\pi {\text{ cm}}^3$, we find that the height $h$ of the cone is 9 cm. Since the radius is 12 cm and the height is 9 cm, the slant height of the cone, which is the same as the distance from $B$ to $C$, is $\sqrt{9^2+12^2}=15$ centimeters. The length of major arc $AC$ is equal to the circumference of the cone, which is $2\pi (12\text{ cm})=24\pi$ cm. The distance all the way around the circle is $2\pi (BC)=30\pi$ cm. Therefore, the central angle of major arc $AC$ measures $\left(\frac{24\pi \text{ cm}}{30\pi \text{ cm}}\right)360^{\circ }=288$ degrees. The measure of angle $ABC$ is $360^{\circ }-288^{\circ }=\boxed{72}$ degrees.