If in `DeltaABC and DeltaDEF` , `(AB)/(DE)=(BC)/(FD)`, then they will be similar, when

Options

A. `angleB=angleE`
B. `angleA=angleD`
C. `angleB=angleD`
D. `angleA=angleF`

Explanation

To determine when triangles ABC and DEF are similar given that the ratios of two pairs of corresponding sides are equal, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
We are given that \(\frac{AB}{DE} = \frac{BC}{FD}\). This indicates that two pairs of corresponding sides of triangles ABC and DEF are proportional.

2. **Identify the Criteria for Similarity**:
There are three main criteria for the similarity of triangles:
- AA (Angle-Angle): If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
- SSS (Side-Side-Side): If the corresponding sides of two triangles are in proportion, the triangles are similar.
- SAS (Side-Angle-Side): If two sides of one triangle are in proportion to two sides of another triangle and the included angles are equal, the triangles are similar.

3. **Apply the Given Ratio**:
Since we know that \(\frac{AB}{DE} = \frac{BC}{FD}\), we can use the SAS criterion for similarity. However, we need to ensure that the angles between these sides are equal.

4. **Check for Angle Equality**:
We need to check if the angle between the sides \(AB\) and \(BC\) (which is angle \(B\)) is equal to the angle between the sides \(DE\) and \(FD\) (which is angle \(D\)). If \( \angle B = \angle D\), then we can conclude that the triangles are similar.

5. **Conclusion**:
Therefore, triangles ABC and DEF will be similar if \(\angle B = \angle D\) in addition to the given proportionality of the sides. This can be stated as:
\[ \text{If } \frac{AB}{DE} = \frac{BC}{FD} \text{ and } \angle B = \angle D, \text{ then } \triangle ABC \sim \triangle DEF \text{ (by SAS criterion)}. \]

### Final Answer:
Triangles ABC and DEF will be similar if \(\angle B = \angle D\).

---