The image shows a geometric figure with parallel lines and intersecting transversals. We are tasked with finding the values of $w$, $x$, $y$, and $z$ based on the given angles of 78° and 82°. Here's the step-by-step breakdown:

Observations:

• The two horizontal lines appear to be parallel, and the angles given (78° and 82°) are formed where transversals intersect these parallel lines.
• The following relationships can be used:
  1. Corresponding angles: When a transversal cuts two parallel lines, corresponding angles are equal.
  2. Alternate interior angles: They are equal when formed between two parallel lines and a transversal.
  3. Linear pair: Angles on a straight line sum to 180°.

Step-by-step solution:

1. Find $w$:

   • $w$ and 78° form a linear pair on the left side.
   • Therefore, $w + 78° = 180°$.
   • Solving for $w$: $w = 180° - 78° = 102°$

2. Find $x$:

   • $x$ and $w$ are vertically opposite angles, so they are equal.
   • Therefore, $x = w = 102°$.

3. Find $y$:

   • $y$ and 82° form a linear pair on the right side.
   • Therefore, $y + 82° = 180°$.
   • Solving for $y$: $y = 180° - 82° = 98°$

4. Find $z$:

   • $z$ and $y$ are vertically opposite angles, so they are equal.
   • Therefore, $z = y = 98°$.

Summary of Results:

• $w = 102°$
• $x = 102°$
• $y = 98°$
• $z = 98°$

Would you like more details or clarification on any step?

Here are five related questions to expand your understanding:

1. How can we prove that corresponding angles are equal when a transversal cuts parallel lines?
2. What is the relationship between alternate interior angles in this configuration?
3. How do linear pairs help in solving for unknown angles in intersecting lines?
4. Can we apply the same method if the lines were not parallel?
5. What other geometric properties can be explored using transversals and parallel lines?

Tip: Always look for linear pairs and vertically opposite angles when solving problems involving intersecting lines.