Step 1: Problem Definition:
Given triangles \( \triangle ABC \) and \( \triangle PQR \) with the ratio \( \frac{AB}{QR} = \frac{BC}{PR} \), determine the conditions for their similarity.
Step 2: Similarity Criteria:
Two triangles are similar if their corresponding sides are proportional and their corresponding angles are equal. This is often referred to as the Side-Side-Angle (SSA) similarity criterion when dealing with two sides and a non-included angle. However, for overall similarity, we require proportionality of all three sides and equality of all three corresponding angles.
We are given that two sides are proportional: \( \frac{AB}{QR} = \frac{BC}{PR} \). For similarity, the third sides must also be proportional, and the included angles must be equal if we were using SAS, or corresponding angles must be equal for AAA or SSS.
Step 3: Applying Similarity Conditions:
For \( \triangle ABC \) and \( \triangle PQR \) to be similar, the ratio of the third pair of corresponding sides must also be equal to the given ratios:\[\frac{AC}{PQ} = \frac{AB}{QR} = \frac{BC}{PR}\]Furthermore, the corresponding angles must be equal. Given the proportional sides \(AB, BC\) and \(QR, PR\), similarity implies that the angle opposite to \(AC\) in \( \triangle ABC \) must be equal to the angle opposite to \(PQ\) in \( \triangle PQR \), and the angle between \(AB\) and \(BC\) must be equal to the angle between \(QR\) and \(PR\). However, the problem statement, by providing \( \frac{AB}{QR} = \frac{BC}{PR} \), suggests a potential link to the SSA case. For similarity in general, we need AAA, SSS, or SAS. If we are given two sides proportional, we need either the included angle to be equal (SAS) or all angles to be equal (AAA). In the context of the given ratios, if we assume that \( \angle B \) and \( \angle R \) are the angles included between the given proportional sides, then for similarity by SAS, we would need \( \angle B = \angle R \). If the proportionality is given, and we are aiming for general similarity, we ultimately need all corresponding angles equal.
However, the standard conditions for similarity that arise from the proportionality of sides are SSS and SAS. If \( \frac{AB}{QR} = \frac{BC}{PR} \) is given, and we assume it implies similarity, it's most likely referring to a scenario where these sides and an included angle lead to similarity. The most comprehensive condition for similarity using side proportions is SSS. If we add the condition \( \angle B = \angle R \), this would align with SAS similarity if \( \angle B \) is between \(AB\) and \(BC\), and \( \angle R \) is between \(QR\) and \(PR\). If we interpret the question to mean similarity based on the given ratios, the most complete set of conditions that ensures similarity, derived from these ratios and potentially an angle, leads to the SSS criterion and equal corresponding angles.
For similarity, the most fundamental conditions are:
1. All corresponding angles are equal: \( \angle A = \angle P, \angle B = \angle Q, \angle C = \angle R \). (AAA similarity)
2. All corresponding sides are proportional: \( \frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR} \). (SSS similarity)
3. Two pairs of corresponding sides are proportional, and the included angles are equal: \( \frac{AB}{PQ} = \frac{BC}{QR} \) and \( \angle B = \angle Q \). (SAS similarity)
Given \( \frac{AB}{QR} = \frac{BC}{PR} \), this implies a specific correspondence. If we are to achieve similarity, we must satisfy one of the above criteria. The condition \( \frac{AB}{QR} = \frac{BC}{PR} \) on its own, without information about angles or the third side, is not sufficient for similarity in general (it could lead to SSA, which is not a similarity criterion unless it's a right triangle and the sides are hypotenuse and leg). However, if the question implies that these ratios, along with an angle condition, lead to similarity, then we need to ensure all sides are proportional and all angles are equal.
The problem statement is slightly ambiguous regarding the exact criterion being tested. If we strictly interpret that similarity is to be achieved from the given ratios, and aiming for a general similarity condition, it suggests that if these ratios hold, AND the third ratio also holds, AND the corresponding angles are equal, then the triangles are similar. The phrasing "when the two triangles will be similar" implies finding sufficient conditions. If we assume the given ratios are part of a larger set of conditions for similarity, then the complete conditions are:
\[\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}\](This assumes a different correspondence than initially stated for clarity on standard SSS). If we stick to the stated correspondence \(\frac{AB}{QR} = \frac{BC}{PR}\), then for similarity (SSS):
\[\frac{AB}{QR} = \frac{BC}{PR} = \frac{AC}{PQ}\]And for angles to be equal, corresponding angles must match. If \( \angle B \) in \( \triangle ABC \) corresponds to \( \angle R \) in \( \triangle PQR \) (which is implied by the side proportionality involving \(AB, BC\) and \(QR, PR\)), then we would need \( \angle B = \angle R \). This would make it SAS similarity if \( \angle B \) is between \(AB\) and \(BC\), and \( \angle R \) is between \(QR\) and \(PR\). If the question is asking for the *conditions for similarity*, and the starting point is \( \frac{AB}{QR} = \frac{BC}{PR} \), the most encompassing conditions for similarity are the full proportionality of sides and equality of angles.
Therefore, the condition for similarity, given the initial proportionality, requires the third side ratio to be equal and the corresponding angles to be equal:
\[\frac{AB}{QR} = \frac{BC}{PR} = \frac{AC}{PQ} \quad \text{and} \quad \angle B = \angle R\](Note: This assumes \( \angle B \) and \( \angle R \) are the corresponding angles based on the side pairings. If \( \angle B \) corresponds to \( \angle Q \), the condition would change). Let's assume the standard correspondence where \( \angle B \) corresponds to \( \angle Q \). However, the given side ratios \( \frac{AB}{QR} = \frac{BC}{PR} \) strongly suggest that \(A \leftrightarrow P, B \leftrightarrow Q, C \leftrightarrow R\) is NOT the correspondence. Instead, it implies a pairing like \(A \leftrightarrow R, B \leftrightarrow Q, C \leftrightarrow P\) or \(A \leftrightarrow P, B \leftrightarrow R, C \leftrightarrow Q\) or some other permutation. If \(AB\) corresponds to \(QR\) and \(BC\) corresponds to \(PR\), then the vertices \(B\) and \(R\) are likely involved in the angle condition. Let's assume, for the sake of providing a concrete answer based on typical problem phrasing, that the intention is for the sides to be proportional in a way that leads to SAS or SSS similarity, and that \( \angle B \) and \( \angle R \) are the corresponding angles. This aligns with the provided solution structure.
Step 4: Conclusion:
Triangles \( \triangle ABC \) and \( \triangle PQR \) are similar if and only if all corresponding sides are proportional and all corresponding angles are equal. Given the initial condition \( \frac{AB}{QR} = \frac{BC}{PR} \), the full conditions for similarity are:\[\boxed{\frac{AB}{QR} = \frac{BC}{PR} = \frac{AC}{PQ} \quad \text{and} \quad \angle B = \angle R}\]