In equal circles, angles have the same ratio as the (ratio of the) circumferences on which they stand, whether they are standing at the centers (of the circles) or at the circumferences. * Let $ABC$ and $DEF$ be equal circles, and let $BGC$ and $EHF$ be angles at their centers, $G$ and $H$ (respectively), and $BAC$ and $EDF$ (angles) at their circumferences. * I say that as circumference $BC$ is to circumference $EF$, so angle $BGC$ (is) to $EHF$, and (angle) $BAC$ to $EDF$.
This is a generalization of Prop 3.26. In congruent circles, the central angles or angles standing upon arcs are proportional if the arcs have proportional lengths.
Proofs: 1