Question

13 paragraph proof given: v is the midpoint of \\( \overline { w x } \\) and \\( \overline { y z } \\). prove: \\( \triangle x v z \cong \triangle w v y \\)

Explanation:

Since \(V\) is the midpoint of \(\overline{WX}\), by the definition of a midpoint, \(WV = XV\). Also, since \(V\) is the midpoint of \(\overline{YZ}\), \(YV=ZV\). The vertical angles \(\angle WVY\) and \(\angle XVZ\) are congruent (vertical angles theorem). In \(\triangle XVZ\) and \(\triangle WYY\), we have \(WV = XV\) (from mid - point), \(YV = ZV\) (from mid - point), and \(\angle WVY=\angle XVZ\) (vertical angles). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle XVZ\cong\triangle WYY\).

Answer:

By the Side - Angle - Side (SAS) congruence criterion, \(\triangle XVZ\cong\triangle WYY\) as \(WV = XV\) ( \(V\) is mid - point of \(WX\) ), \(YV = ZV\) ( \(V\) is mid - point of \(YZ\) ) and \(\angle WVY=\angle XVZ\) (vertical angles).