The diagram shows two similar polygons. Since the polygons are similar, the corresponding sides are proportional, and the corresponding angles are congruent.

We are tasked with finding the values of $w$, $x$, $y$, and $z$.

Step 1: Identifying Corresponding Sides

From the diagram:

• Side corresponding to 8 (left polygon) is 14 (right polygon).
• Side corresponding to 4 (left polygon) is 7 (right polygon).
• Side corresponding to 14 (left polygon) is 21 (right polygon).

Step 2: Establishing the Ratio of Similarity

We can find the ratio of the corresponding sides using one pair of corresponding sides: $\text{Ratio} = \frac{14}{8} = \frac{7}{4}$ Thus, the sides of the second polygon are scaled up by a ratio of $\frac{7}{4}$ compared to the first polygon.

Step 3: Solving for $w$, $x$, $y$, and $z$

Using the ratio $\frac{7}{4}$:

1. Solving for $w$:
   The corresponding side of 4 (right polygon) is $w$ in the left polygon: $w = \frac{4}{7} \times 14 = 8$

2. Solving for $x$:
   The corresponding side of 14 (left polygon) is $x$ in the right polygon: $x = \frac{7}{4} \times 14 = 24.5$

3. Solving for $y$:
   The corresponding side of 14 (left polygon) is $y$ in the right polygon: $y = \frac{7}{4} \times 8 = 14$

4. Solving for $z$:
   The corresponding side of 8 (left polygon) is $z$ in the right polygon: $z = \frac{7}{4} \times 4 = 7$

Final Answers:

• $w = 8$
• $x = 24.5$
• $y = 14$
• $z = 7$

Would you like further details or explanations on any of these steps?

Here are some related questions for further practice:

1. What is the ratio of the areas of the two similar polygons?
2. If another side in the first polygon was 10, what would the corresponding side in the second polygon be?
3. How do you know which sides correspond in similar polygons?
4. What properties of angles can you infer in similar polygons?
5. How would the ratio change if the sides of the larger polygon were reduced by half?

Tip: Always verify that all corresponding angles are congruent when determining similarity of polygons.