The diagram shows two parallel lines $m \parallel n$ cut by a transversal. We are asked to find the values of $x^\circ$ and $y^\circ$.

Step-by-step solution:

1. Angle $y^\circ$:

   • The given angle of $116^\circ$ is vertically opposite to $y^\circ$ since they are on opposite sides of the transversal.
   • Vertically opposite angles are equal, so: $y = 116^\circ$

2. Angle $x^\circ$:

   • The angles $x^\circ$ and $116^\circ$ are on the same side of the transversal and form a co-interior angle pair between the parallel lines $m$ and $n$.
   • For co-interior angles, the sum of the angles is $180^\circ$: $x + 116^\circ = 180^\circ$
   • Solving for $x$: $x = 180^\circ - 116^\circ = 64^\circ$

Final values:

• $x = 64^\circ$
• $y = 116^\circ$

Would you like more details, or do you have any questions about this solution?

Here are 5 related questions to explore:

1. What are corresponding angles in parallel lines, and how do they relate?
2. How can alternate interior angles help in finding unknown angles?
3. What are supplementary angles, and how are they used in geometry?
4. How would the solution change if the transversal formed different angles?
5. How do the properties of parallel lines help in solving real-life geometry problems?

Tip: Always remember that vertically opposite angles are equal when two lines intersect, and co-interior angles on parallel lines sum to 180°.