Let's analyze the diagram and the given information to solve for angle $r$.

We are provided with the following:

• $\angle PQR = 125^\circ$
• $\angle QRS = r$ (This is the unknown we are asked to find)
• $\angle RST = 80^\circ$
• $\angle STU = 44^\circ$

Step 1: Identify the straight-line properties

From the diagram, points $P$, $Q$, and $R$ appear to lie on a straight line, and points $S$, $R$, and $T$ seem to form another line. So, we can use the straight angle property, which tells us that the sum of the angles on a straight line is $180^\circ$.

Step 2: Calculate $r$ using angles on a straight line

We know that: \angle PQR + \angle QRS = 180^\circ \quad \text{(since PQR forms a straight line)} Substitute the given value of $\angle PQR = 125^\circ$: $125^\circ + r = 180^\circ$ Solving for $r$: $r = 180^\circ - 125^\circ = 55^\circ$

Thus, the value of $r$ is $55^\circ$.

Would you like more details, or do you have any questions about this? Here are some related questions:

1. How do we know that $\angle PQR$ and $\angle QRS$ form a straight line?
2. What is the property of angles on a straight line?
3. How would you apply the same reasoning to different geometric configurations?
4. Could other angle properties, like alternate interior angles, apply here?
5. How does this calculation change if a different angle is altered?

Tip: When solving for unknown angles, always check if you can apply the property of angles on a straight line or around a point (both of which sum to 180° and 360° respectively).