Question

complete the proof that \\( \triangle rtu \cong \triangle trs \\).

Explanation:

Step1: Identify the common side

In triangles \( \triangle RTU\) and \( \triangle RTS\), \( \overline{RT}\) is a common side. So, \( \overline{RT}\cong\overline{RT}\) by the Reflexive Property of Congruence.

Step2: Determine the congruence criterion

We are given \( \angle RTS\cong\angle TRU\) (Angle) and \( \angle SRT\cong\angle RTU\) (Angle) and \( \overline{RT}\cong\overline{RT}\) (Side). So, by the Angle - Side - Angle (ASA) congruence criterion, \( \triangle RTU\cong\triangle RTS\).

Answer:

1. \( \angle RTS\cong\angle TRU\) (Given)
2. \( \angle SRT\cong\angle RTU\) (Given)
3. \( \overline{RT}\cong\overline{RT}\) (Reflexive Property of Congruence)
4. \( \triangle RTU\cong\triangle RTS\) (ASA)