The image shows two intersecting lines forming vertical angles, with the angles labeled as follows:

• $z^\circ$
• $(8x + 44)^\circ$
• $(5x + 71)^\circ$

From geometry, we know two things:

1. Vertical angles are equal.

   • So, $z^\circ = (8x + 44)^\circ$ and $(5x + 71)^\circ = (8x + 44)^\circ$.

2. Adjacent angles are supplementary, meaning they sum to 180°.

Let's solve this step-by-step.

Step 1: Set up the equation for the adjacent angles:

$(5x + 71)^\circ + (8x + 44)^\circ = 180^\circ$

Step 2: Solve for $x$:

$5x + 71 + 8x + 44 = 180$ $13x + 115 = 180$ $13x = 180 - 115$ $13x = 65$ $x = \frac{65}{13} = 5$

Step 3: Find $z$:

Since $z = (8x + 44)^\circ$, substitute $x = 5$ into the equation: $z = 8(5) + 44 = 40 + 44 = 84^\circ$

Final Answers:

• $x = 5$
• $z = 84^\circ$

Would you like a more detailed explanation, or do you have any questions?

Here are 5 related questions to explore:

1. How would the solution change if the given angles were part of a triangle?
2. What if the lines were parallel—how would alternate interior angles apply?
3. Can you find the value of $x$ if the adjacent angles were instead equal?
4. How are supplementary and complementary angles different?
5. What geometric properties help us determine the sum of angles in any polygon?

Tip: Vertical angles are always congruent, making them a key property in many geometry problems!