Check the picture below.
about ∡QML
well, QM is a diameter chord, and LM is a tangent line, the point of tangency is always a right-angle, as you see it in darkred there.
about ∡PMN
well, first off, let's notice the inscribed angle of 27°, by the inscribed angle theorem, that means the intercepted arc is twice that much, as you see it there in green.
QM is a diameter, so the arcQPM is 180°, if we subtract the 54° from arcPQ, we're left with the arcPM you see there of 126°.
by the tangent-chord angle theorem, ∡PMN is half of arcPM.
