To establish that triangles ΔOPQ and ΔRST are congruent by the Side-Side-Side (SSS) congruence theorem, you need to show that all three sides of one triangle are equal to the corresponding three sides of the other triangle.
Given the pairs of sides for the triangles:
- For triangle ΔOPQ, you have sides OQ, OP, and PQ.
- For triangle ΔRST, you have sides RS, RT, and ST.
From the options provided, the relevant piece of information needed to complete the SSS condition would be the lengths of all three sides. Since the question asks for a single piece of information needed, you would look for the option that allows you to establish equality for a side in one triangle to a side in the other triangle.
Let's analyze the potential options:
- QO ≅ TR: This tells us that the side QO is equal to TR, which is not sufficient because you still have two more sides left to establish to prove SSS.
- PQ ≅ RT: This pairs one side of ΔOPQ with one side of ΔRST but does not help with the other two sides.
- OQ ≅ RS: Similar to the previous option; this connects one side but not enough for all three.
- OP ≅ RT: This also connects one side but not enough for SSS confirmation.
The missing piece to effectively apply SSS must connect the remaining side of one triangle with a side of the other triangle.
Since we can infer that none of the options listed can independently satisfy the SSS criteria to prove congruence without establishing all three pairs of matching sides, we ideally need to know the length of a side in the second triangle that corresponds to either of the remaining sides of the first triangle.
In a more practical situation, one would actually need all three sides established. However, if we had to select the best option from the provided choices to assert progress towards SSS congruence, one would look for an option that aligns the non-paired sides more effectively with the potential equality of sides already given.
Therefore, if the question specifically expects one additional piece of information, you could argue that ensuring either configuration that connects the adequately matched sides would be essential. Without more context, you can select the odds with potential algebraic match:
If forced to choose among a set of incomplete information, an indication that either OQ matches RS or any two remaining side measures are known would progress you further toward congruence transitively satisfying both conditions needed, preferably leaning towards a complete measure.
Given the options presently does not yield a satisfactory completion without full coupling, you cannot conclusively claim with certainty solely on what is offered to present congruence by SSS therapy given the sufficient lack of matching sets from merely the information outlined. If additional triangulations need emphasis beyond recognition, each distinct pair must manifest across triangles concurrently corroboratively into a match.
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