A picture is shown. Within the picture, there are two triangles shown. One is labeled OPQ, and the other RST. On the line between angles P and Q, there are two dashes on it. On the line between O and P, there is one dash on it. On the line between R and S, there is one dash through it. And on the line between T and S, there are once again, two dashes through it.

Question

A picture is shown. Within the picture, there are two triangles shown. One is labeled OPQ, and the other RST. On the line between angles P and Q, there are two dashes on it. On the line between O and P, there is one dash on it. On the line between R and S, there is one dash through it. And on the line between T and S, there are once again, two dashes through it.
What additional piece of information is needed in order to state that △OPQ≅△RST by the SSS Congruence Theorem? (1 point) Responses PQ¯¯¯¯¯¯¯¯≅RT¯¯¯¯¯¯¯ Modifying above upper P upper Q with bar congruent to Modifying above upper R upper T with bar OP¯¯¯¯¯¯¯¯≅RT¯¯¯¯¯¯¯ Modifying above upper O upper P with bar congruent to Modifying above upper R upper T with bar OQ¯¯¯¯¯¯¯¯≅RS¯¯¯¯¯¯¯ Modifying above upper O upper Q with bar congruent to Modifying above upper R upper S with bar QO¯¯¯¯¯¯¯¯≅TR¯¯¯¯¯¯¯

GPT-4o mini · Answer

To determine that triangles $ \triangle OPQ $ and $ \triangle RST $ are congruent using the SSS (Side-Side-Side) Congruence Theorem, we need to show that all three corresponding sides of the triangles are congruent. From the information provided in your description, we know that: - The sides $ PQ $ and $ TS $ are marked with two dashes (indicating they are congruent). - The sides $ OP $ and $ RS $ are marked with one dash (indicating they are congruent). However, to use the SSS theorem, we also need to know the length of the third pair of corresponding sides: - The triangle $ OPQ $ has sides $ OP $, $ PQ $, and $ OQ $. - The triangle $ RST $ has sides $ RS $, $ RT $, and $ ST $. Since the sides $ OP $ and $ RS $ are already congruent (one dash), and $ PQ $ and $ ST $ are congruent (two dashes), we need information about one of the remaining sides to apply SSS properly. Thus, you need to know either: - $ OQ $ must be congruent to $ RT $, or - $ OQ $ must be congruent to $ TR $, or - $ OQ $ must be congruent to $ RS $. Since the most appropriate choice from your options would be relating the third side, we would conclude that: **We need:** $ OQ \cong RT $ (as this directly corresponds to completing the SSS criteria). Based on your options, the response would be: **$ OQ \cong RS $**