If in two triangles ABC and PQR, `(AB)/(PQ) = (BC)/(RP)`, then for the two triangles to be similar, which of the following condition is necessary?

Options

A. `angleB = angleQ`
B. `angleA = angleP`
C. `angleB = angleP`
D. `angleA = angleQ`

Answer

To determine the necessary condition for triangles ABC and PQR to be similar given that \(\frac{AB}{PQ} = \frac{BC}{RP}\), we can use the Side-Angle-Side (SAS) similarity criterion. Here’s the step-by-step solution:

### Step-by-Step Solution:

1. **Identify the Given Proportions**:
We are given that:
\[ \frac{AB}{PQ} = \frac{BC}{RP} \]
This indicates that two sides of triangle ABC are proportional to two sides of triangle PQR.

2. **Understanding the SAS Similarity Criterion**:
According to the SAS similarity criterion, two triangles are similar if:
- Two sides of one triangle are in proportion to two sides of another triangle.
- The angle included between those two sides is equal.

3. **Identify the Included Angle**:
In triangle ABC, the sides AB and BC are given. The angle included between these two sides is angle B.
In triangle PQR, the corresponding sides are PQ and RP, and the included angle between these two sides is angle P.

4. **Set the Condition for Similarity**:
For triangles ABC and PQR to be similar, the included angles must also be equal:
\[ \angle B = \angle P \]

5. **Conclusion**:
Therefore, the necessary condition for triangles ABC and PQR to be similar, given the proportional sides, is:
\[ \text{If } \angle B = \angle P, \text{ then triangles ABC and PQR are similar.} \]

### Final Answer:
The necessary condition for the triangles ABC and PQR to be similar is that \(\angle B = \angle P\).

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