The problem asks us to find $x$ given the following:

• $m \angle ZIJ = -4 + 5x$
• $m \angle HIJ = 176^\circ$
• $m \angle HIZ = 12x + 10$

Since $\angle ZIJ$, $\angle HIJ$, and $\angle HIZ$ are consecutive angles around the point $I$, we can apply the angle sum property. These three angles together must add up to form a straight line, which is $180^\circ$.

So, we have the equation:

$m \angle ZIJ + m \angle HIJ + m \angle HIZ = 180^\circ$

Substitute the given expressions for the angles:

$(-4 + 5x) + 176 + (12x + 10) = 180$

Now, solve for $x$:

$5x - 4 + 176 + 12x + 10 = 180$

Simplify:

$17x + 182 = 180$

Subtract 182 from both sides:

$17x = -2$

Now, divide by 17:

$x = -\frac{2}{17}$

Thus, $x = -\frac{2}{17}$.

Would you like further clarification or more details on any step?

Here are 5 related questions for practice:

1. How would you solve a similar problem if the angle sum was different?
2. What are supplementary angles, and how do they apply here?
3. Can you identify when angles are complementary instead of supplementary?
4. How does the geometry of the figure help in setting up the equation?
5. What would happen if the angles were given in radians instead of degrees?

Tip: Always simplify the given equations as much as possible before solving for unknowns to reduce errors.