Question

5.) given: ∠zxw ≅ ∠ywx and ∠zwx ≅ ∠yxw prove: (overline{zw}congoverline{yx})

Explanation:

Step1: Identify common side

In \(\triangle ZWX\) and \(\triangle YXW\), \(\overline{WX}\) is common, so \(\overline{WX}\cong\overline{WX}\) (Reflexive property of congruence).

Step2: List angle - angle - side (AAS) conditions

We are given \(\angle ZXW\cong\angle YWX\) and \(\angle ZWX\cong\angle YXW\), and from Step 1 \(\overline{WX}\cong\overline{WX}\). So, by the Angle - Angle - Side (AAS) congruence criterion, \(\triangle ZWX\cong\triangle YXW\).

Step3: Use corresponding parts of congruent triangles

Since \(\triangle ZWX\cong\triangle YXW\), the corresponding parts are congruent. So, \(\overline{ZW}\cong\overline{YX}\) (Corresponding parts of congruent triangles are congruent, CPCTC).

Answer:

\(\overline{ZW}\cong\overline{YX}\) is proved by showing \(\triangle ZWX\cong\triangle YXW\) using AAS and then using CPCTC.