Question

z is the midpoint of \\( \overline{xy} \\) and \\( \overline{xy} \perp \overline{wz} \\). complete the proof that \\( \angle ywz \cong \angle xwz \\).

Explanation:

Step1: Use the definition of mid - point

Since \(Z\) is the mid - point of \(\overline{XY}\), by the definition of mid - point, \(XZ = YZ\). So, \(\overline{XZ}\cong\overline{YZ}\) (Statement 4).

Step2: Use the property of perpendicular lines

Given \(\overline{XY}\perp\overline{WZ}\), then \(\angle WZX=\angle WZY = 90^{\circ}\). So, \(\angle WZX\cong\angle WZY\) (Statement 3).

Step3: Use the reflexive property

\(\overline{WZ}\cong\overline{WZ}\) (Statement 5) by the reflexive property of congruence (\(a = a\) for any segment \(a\)).

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

In \(\triangle WXZ\) and \(\triangle WYZ\), we have \(\overline{XZ}\cong\overline{YZ}\) (from Step 1), \(\angle WZX\cong\angle WZY\) (from Step 2), and \(\overline{WZ}\cong\overline{WZ}\) (from Step 3). So, \(\triangle WXZ\cong\triangle WYZ\) (Statement 6) by the SAS congruence criterion (\(SAS\): if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent).

Step5: Use the Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

Since \(\triangle WXZ\cong\triangle WYZ\), then \(\angle YWZ\cong\angle XWZ\) (Statement 7) by CPCTC.

Answer:

The proof is completed by using the definition of mid - point, properties of perpendicular lines, reflexive property, \(SAS\) congruence criterion, and \(CPCTC\).