We are given the following information from the diagram and problem:

1. $AB \parallel CD$,
2. $\angle PNO = 90^\circ$,
3. $\angle OLM = 120^\circ$,
4. We are asked to find $\angle PQR$.

Step-by-Step Solution:

1. Identifying the Angles

• Since $AB \parallel CD$ and $PQ$ is a transversal, the corresponding angles formed on the parallel lines can help us relate $\angle PQR$ and other angles in the diagram.

2. Utilizing the Given Angles

• $\angle PNO = 90^\circ$ indicates that line $NO$ is perpendicular to $PN$.
• $\angle OLM = 120^\circ$, and $L$ is where the lines meet. Since $AB \parallel CD$, we can utilize this angle in relation to $PQR$.

3. Exterior Angle Theorem

Using the exterior angle theorem:

• The exterior angle of a triangle is equal to the sum of the two opposite interior angles. Here, $\angle OLM$ is an exterior angle of triangle $PQR$. Therefore, $\angle PQR + \angle OLM = 180^\circ$ Since $\angle OLM = 120^\circ$, we get: $\angle PQR + 120^\circ = 180^\circ$ Solving for $\angle PQR$: $\angle PQR = 180^\circ - 120^\circ = 60^\circ$

Thus, the required angle $\angle PQR = 60^\circ$.

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Additional Questions:

1. How do parallel lines help in calculating angles in a transversal?
2. What is the significance of the exterior angle theorem in triangle geometry?
3. Can you explain the concept of corresponding angles in parallel line transversals?
4. How do perpendicular lines affect the calculations in geometric problems?
5. What are some other ways to calculate angles in geometric figures?

Tip:

When dealing with parallel lines and transversals, always look for corresponding, alternate interior, or exterior angles, as they often provide critical relationships.