Let's solve this step by step:

1. Angle Relationships: The diagram involves a straight line and a triangle. The angles form supplementary and triangle angle relationships.

2. Step 1: Find $x$:

   • On the straight line, $128^\circ + x = 180^\circ$ (angles on a straight line add up to $180^\circ$).
   • Solve for $x$: $x = 180^\circ - 128^\circ = 52^\circ$

3. Step 2: Find $y$:

   • In the triangle, the angles add up to $180^\circ$. The triangle has angles $x$, $y$, and $83^\circ$.
   • Using the calculated value of $x = 52^\circ$: $x + y + 83^\circ = 180^\circ$
   • Substitute $x = 52^\circ$: $52^\circ + y + 83^\circ = 180^\circ$
   • Solve for $y$: $y = 180^\circ - 135^\circ = 45^\circ$

Final Answers:

• $x = 52^\circ$
• $y = 45^\circ$

Would you like more details on the steps? Here are some related questions to deepen your understanding:

1. What are supplementary angles, and why do they add up to $180^\circ$?
2. How do we confirm that the sum of angles in a triangle is always $180^\circ$?
3. Can this problem be solved using any geometric theorems like the linear pair theorem?
4. What happens to angle relationships if the triangle is not within a straight-line setup?
5. How would this problem change if the triangle included a right angle?

Tip: Always verify that the sum of the triangle's angles is $180^\circ$ to check your calculations!