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Problem Analysis:

We are given a triangle \( \triangle ABC \) with side lengths:
- \( AB = x + 2 \)
- \( BC = x - 3 \)
- \( AC = y - 1 \)

The task is to determine which angle in \( \triangle ABC \) is the largest. Recall that in any triangle, the largest angle is opposite the longest side. Therefore, we need to compare the side lengths to identify the longest side.

Step-by-Step Solution:

#### Statement (1): \( x - y = 1 \)

1. Express \( y \) in terms of \( x \):
\[
x - y = 1 \implies y = x - 1
\]

2. Substitute \( y = x - 1 \) into the side lengths:
- \( AB = x + 2 \)
- \( BC = x - 3 \)
- \( AC = y - 1 = (x - 1) - 1 = x - 2 \)

3. Compare the side lengths:
- \( AB = x + 2 \)
- \( BC = x - 3 \)
- \( AC = x - 2 \)

To determine which side is the longest, we compare these expressions:
- \( AB = x + 2 \)
- \( AC = x - 2 \)
- \( BC = x - 3 \)

Clearly, \( AB = x + 2 \) is the largest because:
\[
x + 2 > x - 2 > x - 3
\]

4. Conclusion from Statement (1):
- The longest side is \( AB \).
- Therefore, the largest angle is opposite \( AB \), which is \( \angle C \).

Statement (1) is sufficient.

#### Statement (2): \( y = 8 \)

1. Substitute \( y = 8 \) into the side lengths:
- \( AB = x + 2 \)
- \( BC = x - 3 \)
- \( AC = y - 1 = 8 - 1 = 7 \)

2. Compare the side lengths:
- \( AB = x + 2 \)
- \( BC = x - 3 \)
- \( AC = 7 \)

To determine which side is the longest, we need to compare \( AB \), \( BC \), and \( AC \). However, without knowing the value of \( x \), we cannot definitively say which side is the longest. For example:
- If \( x = 10 \):
\[
AB = 10 + 2 = 12, \quad BC = 10 - 3 = 7, \quad AC = 7
\]
Here, \( AB \) is the longest.
- If \( x = 5 \):
\[
AB = 5 + 2 = 7, \quad BC = 5 - 3 = 2, \quad AC = 7
\]
Here, \( AB \) and \( AC \) are equal, and both are longer than \( BC \).

Since the comparison depends on the value of \( x \), we cannot determine the longest side uniquely.

3. Conclusion from Statement (2):
- Statement (2) is insufficient because it does not provide enough information to determine the longest side.

#### Final Answer:
- Statement (1) is sufficient.
- Statement (2) is insufficient.

\[
\boxed{A}
\]