Question

given: \\(\overline{qr} \perp \overline{pt}\\) and \\(\angle qpr \cong \angle str\\)
prove: \\(\triangle pqr \sim \triangle tsr\\)
assemble the proof by dragging tiles to the statements and reasons columns.

Explanation:

Step1: Identify Right Angles

Since \( \overline{QR} \perp \overline{PT} \), \( \angle QRP = 90^\circ \) and \( \angle SRT = 90^\circ \). So, \( \angle QRP \cong \angle SRT \) (all right angles are congruent).

Step2: Use Given Angle Congruence

We are given \( \angle QPR \cong \angle STR \).

Step3: Apply AA Similarity Criterion

In \( \triangle PQR \) and \( \triangle TSR \), we have two pairs of congruent angles: \( \angle QPR \cong \angle STR \) and \( \angle QRP \cong \angle SRT \). By the Angle - Angle (AA) similarity postulate, if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. So, \( \triangle PQR \sim \triangle TSR \).

Answer:

To prove \( \triangle PQR \sim \triangle TSR \):

1. Statement 1: \( \overline{QR} \perp \overline{PT} \) (Given)

• Reason 1: Given
• Implication: \( \angle QRP = 90^\circ \) and \( \angle SRT = 90^\circ \), so \( \angle QRP \cong \angle SRT \) (All right angles are congruent)

1. Statement 2: \( \angle QPR \cong \angle STR \) (Given)

• Reason 2: Given

1. Statement 3: \( \triangle PQR \sim \triangle TSR \)

• Reason 3: AA (Angle - Angle) Similarity Postulate (If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar)