Question

complete the proof that \\( \triangle stv \cong \triangle ruv \\).

Explanation:

Step1: Analyze the given information

We are given that \(\overline{UV}\cong\overline{TV}\), \(\overline{SV}\cong\overline{RV}\), and \(\angle RVU\cong\angle SVT\) (by the Vertical Angle Theorem).

Step2: Recall the Side - Angle - Side (SAS) congruence criterion

The SAS congruence criterion states that 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.
In \(\triangle STV\) and \(\triangle RUV\), we have:

• Side: \(\overline{SV}\cong\overline{RV}\) (given)
• Angle: \(\angle SVT\cong\angle RVU\) (Vertical Angle Theorem)
• Side: \(\overline{TV}\cong\overline{UV}\) (given)

Answer:

By the Side - Angle - Side (SAS) congruence criterion, \(\triangle STV\cong\triangle RUV\). So the reason for step 4 is "Side - Angle - Side (SAS) Congruence Criterion".