Question

wx | z. complete the proof that ∠wyx ≅ ∠ywz.

1. wx = yz
2. wx | z
3. ∠wyz = ∠xwy
4. wy = wy
5. △wyx ≅ △zyw
6. ∠wyx ≅ ∠ywz

Explanation:

Step1: Given

From the problem statement, we know \(WX = YZ\) (given in the problem context for the proof).

Step2: Given

Also given \(WX\parallel YZ\).

Step3: Alternate - interior angles

Since \(WX\parallel YZ\) and \(WY\) is a transversal, \(\angle ZYW=\angle XWY\) (alternate - interior angles theorem: if two parallel lines are cut by a transversal, then alternate - interior angles are congruent).

Step4: Reflexive property

\(WY = WY\) by the reflexive property of equality (a segment is congruent to itself).

Step5: SAS (Side - Angle - Side)

In \(\triangle WXY\) and \(\triangle YZW\), we have \(WX = YZ\) (from step 1), \(\angle ZYW=\angle XWY\) (from step 3), and \(WY = WY\) (from step 4). So, \(\triangle WXY\cong\triangle YZW\) by the SAS (Side - Angle - Side) congruence criterion.

Step6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle WXY\cong\triangle YZW\), then \(\angle WYX=\angle YWZ\) (CPCTC: if two triangles are congruent, then their corresponding parts are congruent).

Answer:

1. Given
2. Given
3. Alternate - interior angles theorem
4. Reflexive property of equality
5. SAS (Side - Angle - Side) congruence criterion
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)