The diagram presents a quadrilateral $PQRS$ with angles labeled, and we're tasked with finding $x + y$.

Steps to solve:

1. Understanding the Geometry: The figure suggests that lines $PQ$ and $SR$ are parallel, and lines $PS$ and $QR$ are also parallel, forming a parallelogram-like shape.

2. Angles in the Diagram:

   • At $P$: $83^\circ$
   • At $Q$: $105^\circ$
   • At $R$: $105^\circ$
   • At $S$: $x^\circ$

   Additionally, external angles at $P$ and $Q$ are labeled as supplementary to $x^\circ$ and $y^\circ$, respectively.

3. Using the properties of parallel lines:

   • Since $PQ \parallel SR$, angles $P$ and $S$ are supplementary. Therefore, $83^\circ + x^\circ = 180^\circ$ Solving for $x$: $x = 180^\circ - 83^\circ = 97^\circ$

   • Similarly, angles $Q$ and $R$ are supplementary because $PS \parallel QR$. Hence: $105^\circ + y^\circ = 180^\circ$ Solving for $y$: $y = 180^\circ - 105^\circ = 75^\circ$

4. Sum of $x + y$: $x + y = 97^\circ + 75^\circ = 172^\circ$

Conclusion:

The value of $x + y$ is $172^\circ$.

Would you like further details or clarifications on this?

Related Questions:

1. What are the properties of parallelograms regarding angles?
2. Can you explain the concept of supplementary angles in more detail?
3. How do we determine if lines are parallel using angles?
4. What is the role of alternate interior angles in parallel lines?
5. How do external angles relate to internal angles in polygons?

Tip:

Always remember, in parallelograms, opposite angles are equal, and adjacent angles are supplementary.