Let's analyze the problem shown in the image.

Problem Breakdown:

The image depicts two intersecting lines, creating two pairs of vertically opposite angles. The given expressions for these angles are:

• $2x + 8^\circ$
• $3x + 17^\circ$
• $5y + 15^\circ$

Since vertically opposite angles are equal, we can set up the following equations:

1. $2x + 8^\circ = 3x + 17^\circ$ (because the angles opposite to each other must be equal)
2. $3x + 17^\circ = 5y + 15^\circ$

Solving the Equations:

1. Solve for $x$ in the first equation: $2x + 8^\circ = 3x + 17^\circ$ Subtract $2x$ from both sides: $8^\circ = x + 17^\circ$ Subtract $17^\circ$ from both sides: $x = 8^\circ - 17^\circ$ $x = -9^\circ$

2. Now, solve for $y$ using the second equation: $3x + 17^\circ = 5y + 15^\circ$ Substitute $x = -9$: $3(-9) + 17^\circ = 5y + 15^\circ$ Simplify: $-27^\circ + 17^\circ = 5y + 15^\circ$ $-10^\circ = 5y + 15^\circ$ Subtract $15^\circ$ from both sides: $-25^\circ = 5y$ Divide by 5: $y = -5^\circ$

Final Answer:

• $x = -9^\circ$
• $y = -5^\circ$

Would you like further details or have any questions? Here are some related questions to consider:

1. What is the definition of vertically opposite angles?
2. How can you determine the relationship between the angles formed by intersecting lines?
3. If a similar problem included a third variable, how would you approach solving it?
4. What do the negative values of $x$ and $y$ imply about the angles?
5. How can you verify your solutions for $x$ and $y$?

Tip: Remember that vertically opposite angles are always equal, regardless of the values involved.