To solve the problem in the image, let's break it down step by step.

Given Information:

1. $l \parallel m \parallel n$ (three parallel lines).
2. $\angle QPS = 35^\circ$.
3. $\angle QRT = 55^\circ$.
4. We need to find $\angle PQR$.

Step-by-Step Solution:

Step 1: Recognize the geometry.

The diagram shows three parallel lines intersected by transversal lines $PQ$ and $QR$. The goal is to calculate the interior angle $\angle PQR$ formed at point $Q$.

Step 2: Use the sum of angles around point $Q$.

The angles around point $Q$ are: $\angle QPS + \angle PQR + \angle QRT = 180^\circ$ (Since the sum of angles on a straight line is $180^\circ$.)

Step 3: Substitute the known values.

$35^\circ + \angle PQR + 55^\circ = 180^\circ$

Step 4: Simplify the equation.

$\angle PQR = 180^\circ - (35^\circ + 55^\circ)$ $\angle PQR = 180^\circ - 90^\circ$ $\angle PQR = 90^\circ$

Final Answer:

$\boxed{\angle PQR = 90^\circ}$

Would you like a detailed explanation of any step, or do you have further questions?

5 Related Questions:

1. How do you calculate angles formed by parallel lines and transversals?
2. What are alternate interior angles and their properties?
3. If $\angle QPS = 40^\circ$, how does $\angle PQR$ change?
4. Can the value of $\angle PQR$ ever be obtuse in this setup?
5. How do supplementary angles work in geometry?

Tip:

Always remember that the sum of angles around a point on a straight line is $180^\circ$, a fundamental rule in geometry!