Question

1. given: (overline{wx}paralleloverline{yz},overline{wx}congoverline{yz}) prove: (\triangle wxzcong\triangle yzx)

Explanation:

Step1: Identify the congruent sides

Given \( \overline{WX}\cong\overline{YZ}\) (given in the problem statement).

Step2: Identify the common side

\(\overline{ZX}\cong\overline{ZX}\) (reflexive property of congruence: any segment is congruent to itself).

Step3: Use the property of parallel lines to find congruent angles

Since \(WX\parallel YZ\), then \(\angle WXZ\cong\angle YZX\) (alternate - interior angles theorem: if two parallel lines are cut by a transversal, then alternate - interior angles are congruent. Here, the transversal is \(ZX\)).

Step4: Apply the SAS (Side - Angle - Side) congruence criterion

In \(\triangle WXZ\) and \(\triangle YZX\), we have \(\overline{WX}\cong\overline{YZ}\), \(\angle WXZ\cong\angle YZX\), and \(\overline{ZX}\cong\overline{ZX}\). By the SAS congruence criterion, \(\triangle WXZ\cong\triangle YZX\).

Answer:

\(\triangle WXZ\cong\triangle YZX\) by the SAS (Side - Angle - Side) congruence criterion.