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

The first given is \(\angle P\cong\angle S\) and \(\overline{TQ}\cong\overline{RQ}\), so the first statement is \(\angle P\cong\angle S,\overline{TQ}\cong\overline{RQ}\)

Step2: State the reason for \(\angle RQS\cong\angle TQP\)

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

Step3: Fill in the third statement

Since we are using the AAS (Angle - Angle - Side) congruence criterion, and we have two angles (\(\angle P\cong\angle S\) and \(\angle RQS\cong\angle TQP\)) and a side (\(\overline{TQ}\cong\overline{RQ}\)), the third statement is \(\triangle QRS\cong\triangle QTP\)

Answer:

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