Question

alycia tried to prove that $\triangle opq \cong \triangle str$.
image of two triangles with some labeled sides and angles, and a table of statements and reasons for the proof
what is the first error alycia made in her proof?
choose 1 answer:
\\(\circ\\) alycia used an invalid reason to justify the congruence of a pair of sides or angles.
\\(\circ\\) alycia only established some of the necessary conditions for a congruence criterion.
\\(\circ\\) alycia established all necessary conditions, but then used an inappropriate congruence criterion.
\\(\circ\\) alycia used a criterion that does not guarantee congruence.

Explanation:

To determine the first error, we analyze each step:

1. Step 1: \( OP = ST = 4 \) is given, correct.
2. Step 2: The diagram shows \( OP \parallel ST \) (not \( RT \)), so the statement \( OP \parallel RT \) is incorrect, but the "Given" reason is misapplied here? Wait, no—wait, the congruence criterion. Wait, the triangles are \( \triangle OPQ \) and \( \triangle STR \) (or \( \triangle STR \)? Wait, the proof step 4 says \( \triangle OPQ \cong \triangle STR \) by "side - angle congruence"—but there's no such congruence criterion. The valid criteria are SAS, ASA, SSS, AAS, HL. "Side - angle" (if it's SSA) does not guarantee congruence. So Alicia used a criterion (side - angle) that doesn't guarantee congruence. Wait, let's re - check:

• Step 3: \( \angle O \cong \angle S \) (alternate interior angles, but if \( OP \parallel ST \), then transversal \( OS \) gives \( \angle O \cong \angle S \) (alternate interior angles). Then \( OP = ST \) (side), \( \angle O=\angle S \) (angle). But for SAS, we need the included angle. Here, if we have \( OP \) and \( OQ \), and \( ST \) and \( SR \), but the "side - angle" (if it's SSA) is not a valid congruence criterion. So the error is that she used a criterion (side - angle congruence) that does not guarantee congruence. So the correct option is the one that says "Alycia used a criterion that does not guarantee congruence."

Answer:

D. Alycia used a criterion that does not guarantee congruence.