https://file.aiquickdraw.com/mathbot/file/1726258541793q9dyw569.jpg

Let's break down the problem from the image you provided. We have two parts, and I'll address each one:

### Part 1:
#### Problem:
In the diagram, it states that DE is parallel to XY, and we are given:
- \( XD = 14.6 \, \text{cm} \)
- \( ZX = 24.8 \, \text{cm} \)
- \( DE = 6.8 \, \text{cm} \)

We are tasked with naming the similar triangles and finding the measure of \( XY \).

#### Solution:
Since \( DE \parallel XY \), we can conclude that triangles \( \triangle ZDE \) and \( \triangle ZXY \) are **similar** by the **AA similarity criterion** (both share angle \( \angle Z \) and the corresponding angles at points \( D \) and \( X \) are equal because of the parallel lines).

For similar triangles, the corresponding sides are proportional. So, we set up the following proportion based on the sides we are given:
\[
\frac{DE}{XY} = \frac{ZD}{ZX}
\]
Substitute the known values:
\[
\frac{6.8}{XY} = \frac{14.6}{24.8}
\]
Now, solve for \( XY \):
\[
XY = \frac{6.8 \times 24.8}{14.6}
\]
Let me calculate that for you:

\[
XY = \frac{6.8 \times 24.8}{14.6} = 11.56 \, \text{cm}
\]

So, the length of \( XY \) is approximately **11.56 cm**.

### Part 2:
#### Problem:
For triangles \( \triangle ABC \) and \( \triangle STU \), the corresponding sides are proportional because the triangles are similar. You are given:
- \( BC = 30 \, \text{cm} \)
- \( AB = 40 \, \text{cm} \)
- \( TU = 25 \, \text{cm} \)

We are asked to find the length of side \( ST \).

#### Solution:
Using the proportionality of the corresponding sides in similar triangles:
\[
\frac{ST}{AB} = \frac{TU}{BC}
\]
Substitute the known values:
\[
\frac{ST}{40} = \frac{25}{30}
\]
Now, solve for \( ST \):
\[
ST = \frac{40 \times 25}{30} = \frac{1000}{30} = 33.33 \, \text{cm}
\]

Thus, the length of \( ST \) is approximately **33.33 cm**.

---

Would you like more details on either part? Here are some related questions:

1. How do you apply the AA similarity criterion in more complex scenarios?
2. What are other ways to prove triangles are similar?
3. How do you determine which sides correspond when triangles are similar?
4. What if DE and XY weren’t parallel—how would the problem change?
5. Can you solve this using another triangle property or theorem?

**Tip**: When solving similarity problems, always ensure you correctly identify corresponding sides based on the angles or side lengths provided.

In the diagram, DE is parallel to XY, XD = 14.6 cm, ZX = 24.8 cm, and DE = 6.8 cm. Name the similar triangles and find the measure of XY.

Learn how to solve a geometry problem involving similar triangles. Given DE is parallel to XY and side lengths are provided, we use the AA similarity criterion and proportionality to find the length of side XY.

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