The statements you provided seem to be related to proportions and equivalent ratios. Let's break down the propositions:

1. **Given:**
- \( \frac{a}{b} = \frac{c}{d} \)
- \( \frac{a}{d} = \frac{c}{b} \)

2. **Claim:**
- \( a = c \)
- \( b = d \)

### Exploring the Statements:

1. **From** \( \frac{a}{b} = \frac{c}{d} \):
- Cross-multiplying gives us \( ad = bc \).

2. **From** \( \frac{a}{d} = \frac{c}{b} \):
- Cross-multiplying gives us \( ab = cd \).

### Analysis of Claims:
To determine if \( a = c \) and \( b = d \) can be concluded from these equations, we'll analyze them further.

By using the cross-multiplication results:
- If \( ad = bc \) and \( ab = cd \), we can consider the implications.

Let's manipulate both equations:

From \( ad = bc \):
- Rearranging gives \( a = \frac{bc}{d} \).

From \( ab = cd \):
- Rearranging gives \( b = \frac{cd}{a} \).

### Equality Assumptions:
The claim states that \( a = c \) and \( b = d \). We need to check if that leads to consistent results under the given conditions.

Assuming \( a = c \) and \( b = d \):

1. **If \( a = c \):**
- Substitute into \( ad = bc \):
\( a \cdot b = a \cdot d \), which implies \( b = d \).

2. **If \( b = d \):**
- Substitute into \( ab = cd \):
\( a \cdot b = c \cdot b \), which implies \( a = c \).

### Conclusion:
Under the assumptions provided, the conclusion \( a = c \) and \( b = d \) follows logically from the given proportions, assuming both relationships are true.

Thus, the statements are consistent:
- If \( \frac{a}{b} = \frac{c}{d} \) and \( \frac{a}{d} = \frac{c}{b} \), then it logically leads to the conclusion that \( a = c \) and \( b = d \).