Question

Question was provided via image upload.

Explanation:

Identify the given geometric setup

The problem asks which fact is not used to prove that \(\Delta PQR\) is similar to \(\Delta STR\). We are given a larger triangle \(\Delta PQR\) with a segment \(ST\) drawn inside, where \(S\) lies on side \(PR\) and \(T\) lies on side \(QR\).

Analyze the similarity proof

To prove \(\Delta PQR \sim \Delta STR\) using the AA Similarity Postulate:

1. Both triangles share the common angle at vertex \(R\), so \(\angle PRQ \cong \angle SRT\) by the Reflexive Property of Congruence.
2. If segment \(ST\) is parallel to segment \(PQ\) (\(ST \parallel PQ\)), then the corresponding angles are congruent: \(\angle RST \cong \angle RPQ\) and \(\angle RTS \cong \angle RQP\).
3. Therefore, \(\Delta PQR \sim \Delta STR\) by the AA Similarity Postulate (or Triangle Similarity Statements).

Determine the unused fact

Since the multiple-choice options are cut off in the image, we must identify the standard geometric properties that are used versus those that are not used:

• Used facts:
• Reflexive Property of Congruence (to show \(\angle R \cong \angle R\)).
• Corresponding Angles Postulate (to show \(\angle RST \cong \angle RPQ\) or \(\angle RTS \cong \angle RQP\) when \(ST \parallel PQ\)).
• AA Similarity Postulate (to establish similarity).
• Not used facts:
• Any theorems involving perpendicular lines, right angles, or congruence postulates like SAS/SSS congruence, or alternate interior angles (since the angles are corresponding, not alternate interior).

Answer:

Because the multiple-choice options are not visible in the image, the correct option is the one stating a geometric fact that is irrelevant to proving \(\Delta PQR \sim \Delta STR\) via AA similarity.

Typically, in this standard geometry question, the facts used are:

1. The reflexive property of congruence (\(\angle R \cong \angle R\)).
2. Corresponding angles are congruent (\(\angle RST \cong \angle RPQ\) or \(\angle RTS \cong \angle RQP\)) given \(ST \parallel PQ\).
3. The AA Similarity Postulate.

Any option listing a fact other than these (such as Alternate Interior Angles Theorem, Vertical Angles Theorem, or SAS Congruence) is not used to prove the similarity.