Final answer:
Given the provided information, none of the options correctly applies the angle-side-angle (ASA) or side-side-side (SSS) congruence rules due to naming and order inconsistencies. If the conditions for ASA were met, two angles and the included side should be congruent, but no option reflects this properly.
Step-by-step explanation:
The question asks us to determine which triangle congruence rule applies to the given triangles with the provided information. The data suggests that triangle QPR and triangle either RTS or STR have one equal side (QR = SR) and two equal angles (< Q and < S). Since angle-side-angle (ASA) is given, we must have two angles and the included side between them being congruent for the rule to apply.
Therefore, the correct option must match the sides and angles between the triangles accordingly. Here, option B. QPR = STR ; ASA is incorrect because the triangles are not named correctly (triangle names must preserve the order of corresponding vertices). Option A. QPR = RTS ; ASA is incorrect because angles are not named in an order that indicates they are between the sides QR and RS (which are equal). Option C is a repeat of option B and contains the same mistake. Lastly, option D. QPR = RTS ; SSS is incorrect because it suggests side-side-side (SSS) congruence, which would require the knowledge of all three sides being congruent, not just one pair and two angles.
Consequently, none of the options provided is correct based on the information given in the question. There seems to be a typo or mistake within the options. The triangles might be congruent by ASA if the angles and the side mentioned are indeed between those angles, but the correct naming of the triangles to reflect this is not provided in the given options.