In this diagram, we have a pair of intersecting lines forming angles. We can use angle relationships to find the value of $x$ and $y$.

1. Using the linear pair relationship: The angle $(x + 12)^\circ$ and the angle $114^\circ$ are on a straight line, so they form a linear pair. The sum of the angles in a linear pair is $180^\circ$.

   $(x + 12)^\circ + 114^\circ = 180^\circ$ Simplifying this: $x + 12 + 114 = 180$ $x + 126 = 180$ $x = 180 - 126$ $x = 54^\circ$

2. Using the vertical angle relationship: Angle $y^\circ$ is vertically opposite to angle $114^\circ$, and vertically opposite angles are equal. Therefore: $y = 114^\circ$

Thus, the values are:

• $x = 54^\circ$
• $y = 114^\circ$

Let me know if you want further details or have any questions!

Here are 5 related questions to explore further:

1. What are complementary and supplementary angles, and how are they used in geometry?
2. How do parallel lines and transversals create special angle relationships?
3. Can you explain the concept of corresponding angles and their properties?
4. What are alternate interior angles, and how do they relate to parallel lines?
5. How can you determine angle measures in polygons using angle sum properties?

Tip: Always remember that angles on a straight line add up to $180^\circ$, and vertical angles are always congruent!