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Let's break down and solve the problem step by step.

### Part (a) - Corresponding Sides and Ratios:
The two triangles are similar, as indicated by the angle markings. The corresponding sides are:

- The shorter leg of Triangle 1 (15) corresponds to the shorter leg of Triangle 2 (6).
- The longer leg of Triangle 1 (20) corresponds to the longer leg of Triangle 2 (8).

We can now write the ratios of the corresponding sides:

- Ratio of shorter legs: \(\frac{15}{6} = 2.5\)
- Ratio of longer legs: \(\frac{20}{8} = 2.5\)

Thus, the ratio between the corresponding sides is 2.5.

### Part (b) - Hypotenuse Calculation:
Now let's calculate the hypotenuse of each triangle using the Pythagorean theorem:

For Triangle 1:
\[
\text{Hypotenuse}^2 = 15^2 + 20^2 = 225 + 400 = 625
\]
\[
\text{Hypotenuse} = \sqrt{625} = 25
\]

For Triangle 2:
\[
\text{Hypotenuse}^2 = 6^2 + 8^2 = 36 + 64 = 100
\]
\[
\text{Hypotenuse} = \sqrt{100} = 10
\]

The ratio of the hypotenuses is:
\[
\frac{25}{10} = 2.5
\]

The ratio of the hypotenuses is also 2.5, which is consistent with the ratios of the corresponding sides in part (a).

### Part (c) - Transformations:
To map one triangle onto the other, a sequence of transformations could be:
1. **Dilation (Scaling)**: Scale Triangle 2 by a factor of 2.5 to match the size of Triangle 1.
2. **Translation (Shift)**: Translate the scaled Triangle 2 so that its vertices align with Triangle 1.

This sequence will map one triangle onto the other.

Would you like further clarification or details on any part of the solution?

### Follow-up questions:
1. How does the Pythagorean theorem work for finding the hypotenuse in right triangles?
2. Why is the ratio between corresponding sides the same for similar triangles?
3. What is the importance of transformations in geometry?
4. Could you apply these concepts to non-right triangles?
5. How do you determine whether two triangles are similar?

### Tip:
When working with similar triangles, always check if the ratios of corresponding sides are equal to verify their similarity.

The ratios Casey wrote from part (a) of problem 2-58 are ratios of the lengths of corresponding sides of the two shapes. Look at the similar triangles below. Which sides correspond? Write ratios for the corresponding side lengths. Calculate the length of the hypotenuse of each triangle. Is the ratio of the hypotenuses equal to the ratios you found in part (a)? Describe a sequence of transformations that would map one triangle onto the other.

This problem explores the properties of similar triangles, calculating corresponding side ratios and the length of the hypotenuse using the Pythagorean theorem. Additionally, learn how to describe transformations that map one triangle to another. Suitable for Grades 7-9 geometry students.

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