32. Find \(\angle MON\).

In triangle MON, we are given \(\angle M = 105^\circ\) and \(\angle O = 90^\circ\). The sum of angles in a triangle is \(180^\circ\).
So, \(\angle N + \angle M + \angle O = 180^\circ\).
\(\angle N + 105^\circ + 90^\circ = 180^\circ\).
\(\angle N + 195^\circ = 180^\circ\).
This is not possible since the angles must sum to 180.
Let's assume the question requires us to find the measure of angle NOM where N is at the right angle.
\(\angle N + \angle M + \angle O = 180^\circ\).
\(\angle O + 105^\circ + 40^\circ = 180^\circ\).
\(\angle O + 145^\circ = 180^\circ\).
\(\angle O = 180^\circ - 145^\circ\).
\(\angle O = 35^\circ\). Since there is a 90 degree, it means the angle indicated as M is exterior to the triangle itself.
The angles given are at M and P, angle N is 90. Let's assume we need to find angle at M. \(\angle MON = 180 - 90 - (180-105 - 40)\)=180 - 90 - (35)=55, which is incorrect. We will consider the angles as is. So we need to find \(\angle MON\). \(180 - (105+40)=180-145 = 35\)
**Answer: \(\angle MON = 35^\circ\)**