Question

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complete the proof that ( overline{wy} cong overline{wx} ).

Explanation:

Step1: Identify the given angles

The problem states that \( \angle XWZ\cong\angle YWZ\). This is given in the diagram (the two - marked angles at vertex \(W\)).

Step2: Identify the other given angles

\(\angle X\cong\angle Y\) is also given in the diagram (the two - marked angles at vertices \(X\) and \(Y\)).

Step3: Reflexive property of congruence

For segment \(\overline{WZ}\), by the reflexive property of congruence (a segment is congruent to itself), \(\overline{WZ}\cong\overline{WZ}\).

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

Since we have two pairs of congruent angles (\(\angle XWZ\cong\angle YWZ\) and \(\angle X\cong\angle Y\)) and a pair of congruent non - included sides (\(\overline{WZ}\cong\overline{WZ}\)), by the AAS (Angle - Angle - Side) congruence criterion, \(\triangle WXZ\cong\triangle WYZ\).

Step5: Corresponding parts of congruent triangles

If \(\triangle WXZ\cong\triangle WYZ\), then by the property of corresponding parts of congruent triangles (CPCTC), \(\overline{WY}\cong\overline{WX}\).

Answer:

1. Given
2. Given
3. Reflexive property of congruence
4. AAS (Angle - Angle - Side)
5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)