In the image, we have a set of angles formed by intersecting lines. Based on the diagram, we need to use the properties of vertical and linear pairs of angles to solve for $x$ and $y$.

Here are the steps we can take to solve for $x$ and $y$:

1. Equation 1: Vertical Angles

We can observe that the angle $(5x + 4)^\circ$ is vertically opposite to the angle $(3x - 24)^\circ$. Vertically opposite angles are equal, so we can set up the equation: $5x + 4 = 3x - 24$ Solve this equation for $x$.

2. Equation 2: Linear Pair of Angles

The angle $114^\circ$ and $(2y)^\circ$ form a linear pair because they are adjacent and add up to 180°: $114 + 2y = 180$ Solve this equation for $y$.

I will now calculate the values for $x$ and $y$.The solution to the equations is:

• $x = -14$
• $y = 33$

These values satisfy the conditions given by the vertical angles and the linear pair in the diagram.

Would you like more details on the steps? Here are five related questions to consider:

1. How are vertical angles related in geometry?
2. Can linear pairs be formed by any two adjacent angles?
3. What happens if the angles don't sum to 180° in a linear pair?
4. How would these values affect other angles in the diagram?
5. How do we verify the solutions for $x$ and $y$?

Tip: Always check the geometry relationships (like vertical angles or linear pairs) to simplify the problem before diving into algebra.