To solve for $x$ and $z$ in the given diagram, we can apply the following geometric principles:

1. Vertically opposite angles: These angles are equal. In the diagram, the angle labeled $(15x - 91)^\circ$ is vertically opposite the angle labeled $106^\circ$. Therefore, they are equal.

   $15x - 91 = 106$

2. Linear pair: Angles on a straight line add up to $180^\circ$. The angle $z^\circ$ and the $106^\circ$ angle are adjacent angles, forming a linear pair, so:

   $z + 106 = 180$

Step 1: Solve for $x$

Using the equation for vertically opposite angles:

$15x - 91 = 106$

Add $91$ to both sides:

$15x = 106 + 91$ $15x = 197$

Now, divide by $15$:

$x = \frac{197}{15} \approx 13.13$

Step 2: Solve for $z$

Using the equation for the linear pair:

$z + 106 = 180$

Subtract $106$ from both sides:

$z = 180 - 106$ $z = 74$

Final Answers:

• $x \approx 13.13$
• $z = 74$

Would you like further details or explanations? Here are five related questions to explore further:

1. What are supplementary angles, and how do they relate to linear pairs?
2. How can we use the properties of vertically opposite angles in other geometrical problems?
3. If another angle is added, forming a triangle, how would you use angle sum properties to find unknown angles?
4. How does the concept of alternate interior angles apply to parallel lines cut by a transversal?
5. What would change if the given angles were part of a quadrilateral?

Tip: In geometry, always look for relationships like vertically opposite, supplementary, and complementary angles to simplify angle-based problems!