Mike constructed a quadrilateral POQS. He used a compass and straightedge to accurately construct line segment OS, as shown in the figure below:

An angle POQ is drawn. Two similar arcs intersect OP and OQ. Two more similar arcs intersect each other at a point S, and a dashed line is drawn from O to S, such that OS bisects angle POQ.

Which could be the measures of angle POS and angle POQ?