Question

z is the midpoint of \\( \overline{wy} \\) and \\( \overline{vx} \\). complete the proof that \\( \angle x \cong \angle v \\).

Explanation:

Step1: Definition of midpoint

Since \(Z\) is the midpoint of \(\overline{WY}\), by the definition of a midpoint, \(WZ = YZ\).

Step2: Definition of midpoint

Since \(Z\) is the midpoint of \(\overline{VX}\), by the definition of a midpoint, \(VZ=XZ\).

Step3: Vertical angles

\(\angle VZY\) and \(\angle WZX\) are vertical angles. By the vertical - angles theorem, \(\angle VZY\cong\angle WZX\).

Step4: SAS (Side - Angle - Side) congruence

In \(\triangle VYZ\) and \(\triangle WXZ\), we have \(VZ = XZ\) (from step 2), \(\angle VZY=\angle WZX\) (from step 3), and \(YZ = WZ\) (from step 1). So, by the SAS (Side - Angle - Side) congruence criterion, \(\triangle VYZ\cong\triangle WXZ\).

Step5: Corresponding parts of congruent triangles

If \(\triangle VYZ\cong\triangle WXZ\), then by the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem, \(\angle X\cong\angle V\).

Answer:

1. Given
2. Given
3. Definition of midpoint
4. Definition of midpoint
5. Vertical - angles theorem
6. SAS (Side - Angle - Side) congruence
7. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)