Question

If in triangles $ABC$ and $PQR$, $\frac{AB}{QR} = \frac{BC}{PR}$, then they will be similar, when:

• A. $\angle B = \angle Q$
• B. $\angle A = \angle R$
• C. $\angle B = \angle R$
• D. $\angle C = \angle Q$

Answer 1

The Correct Option is C

Step 1: Understand the problem:
We are given two triangles \( \triangle ABC \) and \( \triangle PQR \), and we are provided with the ratio \( \frac{AB}{QR} = \frac{BC}{PR} \). We are asked to determine when the two triangles will be similar.

Step 2: Criteria for similarity of triangles:
Two triangles are similar if their corresponding sides are proportional and their corresponding angles are equal. This is known as the Side-Side-Angle (SSA) criterion for similarity.

In this case, we are given that \( \frac{AB}{QR} = \frac{BC}{PR} \), which means that two sides of the triangles \( \triangle ABC \) and \( \triangle PQR \) are proportional. To ensure that the triangles are similar, the third sides must also be proportional, and the corresponding angles must be equal.

Step 3: Apply the condition for similarity:
For the two triangles \( \triangle ABC \) and \( \triangle PQR \) to be similar, the third sides must also be proportional. That is, we need to check if: \[ \frac{AC}{PQ} = \frac{AB}{QR} = \frac{BC}{PR} \] Additionally, the corresponding angles must be equal: \[ \angle B = \angle R \] Thus, the condition for similarity of the triangles is:
\[ \frac{AB}{QR} = \frac{BC}{PR} = \frac{AC}{PQ} \quad \text{and} \quad \angle B = \angle R \]

Step 4: Conclusion:
The triangles \( \triangle ABC \) and \( \triangle PQR \) will be similar if their corresponding sides are proportional and their corresponding angles are equal. This condition can be stated as:
\[ \boxed{\frac{AB}{QR} = \frac{BC}{PR} = \frac{AC}{PQ} \quad \text{and} \quad \angle B = \angle R} \]