Question

if ( v ) is the midpoint of ( overline{uw} ) and ( overline{tx} ), which of the statements is true?

Explanation:

Step1: Mid - point property

Since \(V\) is the mid - point of \(\overline{UW}\) and \(\overline{TX}\), we have \(UV = WV\) and \(TV=XV\).

Step2: Vertical angles

\(\angle T V U\) and \(\angle X V W\) are vertical angles. By the vertical angles theorem, \(\angle TVU=\angle XVW\).

Step3: SAS congruence criterion

In \(\triangle TUV\) and \(\triangle WXV\), we have \(TV = XV\) (from mid - point), \(\angle TVU=\angle XVW\) (vertical angles), and \(UV = WV\) (from mid - point). So, by the Side - Angle - Side (SAS) congruence criterion, \(\triangle TUV\cong\triangle WXV\).

Answer:

\(\triangle TUV\cong\triangle WXV\) by \(SAS\)