To solve for $x$ and $z$ in this triangle, we will use the property that the sum of the interior angles of a triangle is $180^\circ$.

Given angles:

1. $3x - 2$
2. $2x + 5$
3. $z + 5$

Step 1: Set up the equation

Since the sum of the angles in a triangle is $180^\circ$: $(3x - 2) + (2x + 5) + (z + 5) = 180$

Step 2: Simplify the equation

Combine like terms: $5x + z + 8 = 180$

Now, simplify further: $5x + z = 172$

Step 3: Find a relationship between $x$ and $z$

To solve for $x$ and $z$, we need either an additional relationship or assume some values to analyze. However, this problem may lack a second equation to determine specific values unless further conditions are provided (such as identifying it as an equilateral or isosceles triangle).

Without further information, we cannot uniquely solve for $x$ and $z$ beyond the relationship: $z = 172 - 5x$

Please let me know if there’s additional information about the type of triangle, or if I should assume any specific property, like an isosceles triangle, to proceed with a unique solution.

Would you like more details on the process? Here are some related questions to consider:

1. How would the solution change if the triangle were equilateral?
2. What if it were an isosceles triangle with $3x - 2 = 2x + 5$?
3. Could we solve for specific values if one angle was given explicitly?
4. How do we handle cases where triangles have expressions rather than fixed angles?
5. What are other methods to solve for unknowns in a triangle with algebraic expressions?

Tip: For triangles with variable angles, always check if there’s additional information like side lengths or type (e.g., isosceles) for further constraints.