Let's solve each of the problems step by step, as these involve finding $x$ in similar polygons.

Problem 18:

The two quadrilaterals are similar. For similar figures, corresponding sides are proportional.

We are given:

• One side of the smaller quadrilateral: $x + 5$
• Corresponding side of the larger quadrilateral: $15$
• Another side of the smaller quadrilateral: $4$
• Corresponding side of the larger quadrilateral: $5$

Using the property of similarity: $\frac{x + 5}{15} = \frac{4}{5}$

Simplify: $x + 5 = \frac{4}{5} \cdot 15$ $x + 5 = 12$

Solve for $x$: $x = 12 - 5 = 7$

So, $x = 7$.

Problem 19:

The two trapezoids are similar. Again, corresponding sides are proportional.

We are given:

• One side of the smaller trapezoid: $x + 3$
• Corresponding side of the larger trapezoid: $2x + 2$
• Another side of the smaller trapezoid: $2$
• Corresponding side of the larger trapezoid: $3$

Set up the proportion: $\frac{x + 3}{2x + 2} = \frac{2}{3}$

Cross-multiply: $3(x + 3) = 2(2x + 2)$

Expand both sides: $3x + 9 = 4x + 4$

Solve for $x$: $9 - 4 = 4x - 3x$ $x = 5$

So, $x = 5$.

Problem 20:

The two squares are similar, and their corresponding sides are proportional.

We are given:

• One side of the smaller square: $8x - 1$
• Corresponding side of the larger square: $3x - 1$
• Another side of the smaller square: $12$
• Corresponding side of the larger square: $4$

Set up the proportion: $\frac{8x - 1}{12} = \frac{3x - 1}{4}$

Cross-multiply: $4(8x - 1) = 12(3x - 1)$

Expand both sides: $32x - 4 = 36x - 12$

Solve for $x$: $36x - 32x = 12 - 4$ $4x = 8$ $x = 2$

So, $x = 2$.

Problem 21:

The two polygons are similar.

We are given:

• One side of the smaller polygon: $x + 1$
• Corresponding side of the larger polygon: $8$
• Another side of the smaller polygon: $3x + 1$
• Corresponding side of the larger polygon: $20$

Set up the proportion: $\frac{x + 1}{8} = \frac{3x + 1}{20}$

Cross-multiply: $20(x + 1) = 8(3x + 1)$

Expand both sides: $20x + 20 = 24x + 8$

Solve for $x$: $20 - 8 = 24x - 20x$ $12 = 4x$ $x = 3$

So, $x = 3$.

Final Answers:

1. $x = 7$ (Problem 18)
2. $x = 5$ (Problem 19)
3. $x = 2$ (Problem 20)
4. $x = 3$ (Problem 21)

Do you want a deeper explanation of any step? Here are 5 follow-up questions to expand understanding:

1. Why do corresponding sides of similar polygons maintain the same proportion?
2. How can you check if two polygons are truly similar?
3. What happens if one of the side lengths contains a mistake—how would that impact solving $x$?
4. What other properties of polygons can we determine when they are similar?
5. How do these principles of similarity apply to real-world problems?

Tip:

When working with proportions, always cross-multiply carefully and simplify to avoid errors.