Question

2
given: \\(\overline{pq} \parallel \overline{rs}\\), \\(\angle pqr \cong \angle rsp\\)
prove: \\(\triangle pqr \cong \triangle rsp\\)

statements	reasons
2.	2. alternate interior angles
3. \\(\angle pqr \cong \angle rsp\\)	3.
4. \\(\overline{rp} \cong \overline{pr}\\)	4.
5. \\(\triangle pqr \cong \triangle rsp\\)	5.

Explanation:

Step1: Fill Statement 1

Given is \(\overline{PQ} \parallel \overline{RS}\), \(\angle PQR \cong \angle RSP\), so statement 1 is \(\overline{PQ} \parallel \overline{RS}\), \(\angle PQR \cong \angle RSP\).

Step2: Find Alternate Interior Angles

Since \(\overline{PQ} \parallel \overline{RS}\) and transversal \(\overline{RP}\), alternate interior angles are \(\angle QPR \cong \angle SRP\). So statement 2 is \(\angle QPR \cong \angle SRP\).

Step3: Reason for \(\angle PQR \cong \angle RSP\)

This is given, so reason 3 is "Given".

Step4: Reason for \(\overline{RP} \cong \overline{PR}\)

A segment is congruent to itself (Reflexive Property of Congruence), so reason 4 is "Reflexive Property".

Step5: Prove Triangle Congruence

We have two angles and a non - included side? Wait, no, \(\angle QPR \cong \angle SRP\), \(\angle PQR \cong \angle RSP\), and \(\overline{RP} \cong \overline{PR}\). Wait, actually, using AAS (Angle - Angle - Side) congruence criterion. So reason 5 is "AAS (Angle - Angle - Side) Congruence Criterion" because we have two angles and a non - included side (the side \(\overline{RP}\) is common, but in terms of the triangles \(\triangle PQR\) and \(\triangle RSP\), \(\angle QPR \cong \angle SRP\), \(\angle PQR \cong \angle RSP\), and \(\overline{RP} \cong \overline{PR}\) (which is the side opposite one of the angles? Wait, no, let's re - check. \(\triangle PQR\) has angles \(\angle QPR\), \(\angle PQR\) and side \(\overline{RP}\); \(\triangle RSP\) has angles \(\angle SRP\), \(\angle RSP\) and side \(\overline{PR}\). Since \(\angle QPR \cong \angle SRP\), \(\angle PQR \cong \angle RSP\), and \(\overline{RP} \cong \overline{PR}\), by AAS, \(\triangle PQR \cong \triangle RSP\).

Answer:

1. \(\boldsymbol{\overline{PQ} \parallel \overline{RS}}\), \(\boldsymbol{\angle PQR \cong \angle RSP}\)
2. \(\boldsymbol{\angle QPR \cong \angle SRP}\)
3. Given
4. Reflexive Property (of Congruence)
5. AAS (Angle - Angle - Side) Congruence Criterion