Let's analyze the image and solve for $x$ and $y$ based on the diagram.

Step 1: Identify angle relationships

In the given diagram, we have two intersecting sets of lines forming angles. Based on the angles marked:

• $y^\circ$ is vertically opposite $55^\circ$, so they are equal: $y = 55^\circ$

• $x^\circ$ is vertically opposite $50^\circ$, so they are also equal: $x = 50^\circ$

Solution:

• $y = 55^\circ$
• $x = 50^\circ$

Would you like further explanations on how vertical angles work or any related topics? Here are 5 related questions to deepen your understanding:

1. What is the vertical angle theorem, and how is it applied in geometry?
2. Can you identify other pairs of vertical angles in different diagrams?
3. How do supplementary angles relate to intersecting lines?
4. What other types of angle relationships occur with parallel lines and a transversal?
5. How would you solve for unknown angles if two lines are parallel?

Tip: In any intersecting lines, the opposite (vertical) angles are always equal. This property is useful when solving for unknown angles in geometrical figures.