Given that $QP \parallel MR$ and $LQ \parallel NP$, we are to determine the measure of $\angle QMO$.

To solve this, let's follow the steps carefully:

1. Identify known angles and properties:

   • $\angle LMQ = 75^\circ$
   • $\angle MQR = 45^\circ$
   • $\angle QNP = 22^\circ$

2. Analyze the geometric relationships:

   • Since $QP \parallel MR$ and $LQ \parallel NP$, we can use the properties of parallel lines and transversals.
   • $\angle LMO$ is a corresponding angle to $\angle QMR$, so $\angle LMO = 75^\circ$.
   • $\angle QMO$ forms part of the quadrilateral $LMNO$.

3. Calculate $\angle QMO$:

   • Note that the sum of angles in a quadrilateral is $360^\circ$.
   • $\angle LMN$ and $\angle QNP$ are consecutive angles on a straight line, and hence they add up to $180^\circ$. Since $\angle QNP = 22^\circ$, $\angle LMN = 180^\circ - 22^\circ = 158^\circ$.
   • Given that $\angle LMQ = 75^\circ$ and $\angle QMR = 45^\circ$, the remaining angle $\angle QMO$ in the quadrilateral can be found by: $360^\circ - (\angle LMQ + \angle QMR + \angle LMN) = 360^\circ - (75^\circ + 45^\circ + 158^\circ)$ $360^\circ - 278^\circ = 82^\circ$

Therefore, $\angle QMO = 82^\circ$.

Reasons for the angles:

1. $\angle LMQ = 75^\circ$: Given directly.
2. $\angle QMR = 45^\circ$: Given directly.
3. $\angle LMN = 158^\circ$: Calculated based on supplementary angles.
4. $\angle QMO = 82^\circ$: Calculated as the remaining angle in the quadrilateral.

Would you like further details or have any questions about this solution? Here are some follow-up questions you might consider:

1. How do we determine angles using parallel lines and transversals?
2. What are the properties of angles in a quadrilateral?
3. Can we use the same principles to solve other problems involving parallel lines?
4. How do supplementary and corresponding angles work in geometry?
5. How can we check if our calculated angles are correct?

Tip: Always verify your calculated angles by checking if the total sums to the expected value (e.g., $180^\circ$ for a triangle, $360^\circ$ for a quadrilateral).