If in triangles ABC and PQR, `(AB)/(PQ) = (BC)/(RP)` then write the equality of angles of the two triangles such that two triangles are similar.

Options

A. `angle A = angle `
B. `angle B = angle P`
C. `angle C = angle Q`
D. `angle B = angle Q`

Answer

To determine the equality of angles in triangles ABC and PQR given that \(\frac{AB}{PQ} = \frac{BC}{RP}\), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Given Ratios**:
We are given that \(\frac{AB}{PQ} = \frac{BC}{RP}\). This indicates that the sides of triangle ABC are proportional to the sides of triangle PQR.

2. **Identifying Corresponding Sides**:
From the ratios, we can identify:
- \(AB\) corresponds to \(PQ\)
- \(BC\) corresponds to \(RP\)

3. **Finding the Third Side Ratio**:
Since we have two ratios, we can find the third ratio using the property of similar triangles. For triangles to be similar, the ratio of the third sides must also hold. Therefore, we can write:
\[ \frac{AC}{QR} = k \quad \text{(where \(k\) is some constant)} \]
This means that if \(\frac{AB}{PQ} = k\) and \(\frac{BC}{RP} = k\), then \(\frac{AC}{QR} = k\) must also be true.

4. **Applying the Angle-Angle (AA) Similarity Criterion**:
Since the sides are in proportion, we can conclude that the triangles are similar by the Side-Side-Side (SSS) similarity criterion. This means that the corresponding angles of the triangles are equal.

5. **Writing the Equality of Angles**:
From the similarity of triangles, we can write the following equalities of angles:
- \(\angle A = \angle P\)
- \(\angle B = \angle Q\)
- \(\angle C = \angle R\)

6. **Conclusion**:
Therefore, we can conclude that triangle ABC is similar to triangle PQR, and the equality of angles can be expressed as:
\[ \triangle ABC \sim \triangle PQR \]
with the corresponding angles being:
\(\angle A = \angle P\), \(\angle B = \angle Q\), and \(\angle C = \angle R\).