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?

Unless we were given values of the angles, there is no way to find the numerical value of the angle. However, we can establish a relationship between the measures of angles POS and POQ. Since it was stated that OP and OQ were similar arcs (had the same value), and that OS bisects angle POQ, we can conclude that the two angles (POS and QOS) created from bisecting POQ are each, exactly half the angle POQ. Therefore:

angle POS = 1/2 angle POQ