In the diagram below, prove that: $$\angle QMP=\angle RMP$$ .
You are right about “alternate segment theory” is needed in solving this problem. However, the $O_1$ and $O_2$ are distractions.
Draw the tangent at M and let it cut QPR extended at T. By tangent properties, the blue marked angles are equal (= x, say).
$\angle PMR = x – z$
$= (\angle Q + y) – z$, .... (exterior angle of triangle)
$= y$ .... (because $\angle Q = z$, by “angles in alternate segment”).