Question

1. given: \\( \angle p \cong \angle s, \overline{tq} \cong \overline{rq} \\) prove: \\( \triangle qrs \cong \triangle qtp \\)

Explanation:

Step1: Fill in the first statement

Since the given information includes \(\angle P\cong\angle S\) and \(\overline{TQ}\cong\overline{RQ}\), the first statement should be \(\angle P\cong\angle S,\overline{TQ}\cong\overline{RQ}\)

Step2: Justify \(\angle RQS\cong\angle TQP\)

Vertical angles are congruent. So the reason for \(\angle RQS\cong\angle TQP\) is "Vertical Angles Theorem"

Step3: State the conclusion

By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle QRS\cong\triangle QTP\)

Answer:

1. \(\angle P\cong\angle S,\overline{TQ}\cong\overline{RQ}\)
2. Vertical Angles Theorem
3. \(\triangle QRS\cong\triangle QTP\)