In this problem, we need to find the values of $x$ and $y$ in a triangle with the given angle expressions. Let's go through the solution.

Step 1: Understand the Angles

From the image, we can see that the triangle has three angles:

1. $(3x + 5)^\circ$
2. $(6x - 14)^\circ$
3. $(y + 8)^\circ$

Step 2: Apply the Triangle Sum Property

The sum of the angles in a triangle is always $180^\circ$. Therefore, we can write the equation: $(3x + 5) + (6x - 14) + (y + 8) = 180$

Step 3: Simplify the Equation

Combine like terms: $9x + y - 1 = 180$ Add 1 to both sides to isolate terms: $9x + y = 181$

At this point, we need more information to solve for both $x$ and $y$. However, if there are any additional conditions or relations between the angles (such as if the triangle is a specific type), please share that, and I can proceed accordingly.

Would you like further assistance, or do you have any more details?

Here are 5 related questions that might help deepen your understanding:

1. What happens if two angles in the triangle are equal?
2. How does the exterior angle theorem relate to this problem?
3. Can we apply any trigonometric identities in triangle problems like this?
4. What if the triangle was an isosceles or right triangle?
5. How can we verify if solutions for $x$ and $y$ are valid?

Tip: Remember that in any triangle, the angle sum property ($180^\circ$) is fundamental and always a good starting point for finding unknown values.