What additional piece of information is needed in order to state that OPQ=RST by the SSS congruence theorem?

Question

OP=RT OQ=RS PQ=RT QO=TR

Answer 1

To use the Side-Side-Side (SSS) Congruence Theorem to state that triangles OPQ and RST are congruent (OPQ = RST), we need to establish that all three corresponding sides of the triangles are equal.

From your information, we have:

1. OP = RT
2. OQ = RS
3. PQ = RT

However, we notice that there is an issue with the third equation, as it states that PQ = RT instead of having a corresponding side for PQ in triangle RST.

To conclude that triangles OPQ and RST are congruent by the SSS theorem, we need to confirm that:

• OP = RT (already given)
• OQ = RS (already given)
• PQ = ST (this is the missing information we need)

Thus, the additional piece of information needed is PQ = ST.

Answer 2

PQ=ST is not an option. Choose one of the options provided.

Answer 3

Given the options you provided, we can analyze them:

1. OP = RT (given)
2. OQ = RS (given)
3. PQ = RT (this is incorrect in its corresponding triangle context)
4. QO = TR (this is not a correspondence based on typical labeling)

The correct set of information needed for SSS to apply must involve the lengths of all three sides of both triangles:

• OP = RT (1st side)
• OQ = RS (2nd side)

The next logical option from what you provided that could offer a corresponding length must pertain to the third side of the triangles. Given that there is no option stating PQ = ST or similar, I would choose OP = RT since this ensures the first pair of sides are equal and builds a foundation towards establishing congruence, moving toward utilizing the SSS theorem by confirming the relationships among all sides.

Therefore, if I must choose from the listed options and in the context of what is logically necessary to support SSS, I conclude that one would need:

"OQ = RS" as one of the provided options since it confirms the equality of the second sides relevant to proving congruence.

However, it's crucial to note that all three pairs of sides would ultimately be necessary for using SSS, and that means without having PQ = ST explicitly stated, congruence via SSS cannot be fully established with the provided options.

If required to choose strictly one option from the selected choices provided, it must be: "OQ = RS."