1995 AMC 12 Problem 17

The radii to $A$ and $D$ are perpendicular to the tangent sides

Use the $72^\circ$ exterior angle of a regular pentagon to compare $\overline{AB}$ and $\overline{DC}$

Let $O$ be the circle’s center. The direction changes by $72^\circ$ at each vertex of the pentagon, so the acute angle between lines $AB$ and $DC$ is $36^\circ.$ The radii to their tangency points are perpendicular to these lines; for the circle shown, the minor central angle $\angle AOD$ is the supplementary angle $180^\circ-36^\circ=144^\circ.$ Therefore minor arc $\overset{\frown}{AD}$ measures $144^\circ,$ and the correct answer is E.