The image shows a set of angles formed by two intersecting lines, and we have four expressions representing angles:

1. $(12x + 8)^\circ$
2. $(3x + 17)^\circ$
3. $(5y + 15)^\circ$
4. $\text{Another angle vertically opposite to } (5y + 15)^\circ$.

Since the angles formed by intersecting lines are either vertically opposite or supplementary, we can use the following relationships:

• Vertically opposite angles are equal.
• Supplementary angles add up to $180^\circ$.

Step 1: Use the supplementary relationship

Let's consider the angles on the straight line, which should add up to $180^\circ$. This gives us two equations:

$(12x + 8) + (3x + 17) = 180$ Simplifying: $15x + 25 = 180$ $15x = 155$ $x = \frac{155}{15} = 10.33$

Step 2: Vertically opposite angles

Now, the angles $(5y + 15)^\circ$ and $(3x + 17)^\circ$ are vertically opposite, so they must be equal:

$5y + 15 = 3x + 17$

Substitute the value of $x = 10.33$ into the equation:

$5y + 15 = 3(10.33) + 17$ $5y + 15 = 30.99 + 17$ $5y + 15 = 47.99$ $5y = 47.99 - 15$ $5y = 32.99$ $y = \frac{32.99}{5} = 6.60$

Final values:

• $x \approx 10.33$
• $y \approx 6.60$

Let me know if you'd like further details, clarifications, or calculations!

Here are some follow-up questions to deepen your understanding:

1. What is the relationship between vertically opposite angles and supplementary angles?
2. How can you verify if your solution to $x$ and $y$ satisfies both equations?
3. Can you explore other angle pairs in this diagram to check their consistency?
4. Why do supplementary angles always sum up to $180^\circ$ on a straight line?
5. How can you apply these angle relationships to other geometric figures like triangles or quadrilaterals?

Tip: Always double-check if two angles are vertically opposite or supplementary by looking at their position in the diagram.