Question

1 \\( \angle rts \cong \angle tru \\) 2 \\( \angle srt \cong \angle rtu \\) 3 \\( \overline { rt } \cong \overline { rt } \\) 4 \\( \triangle rtu \cong \triangle trs \\)

Explanation:

Step1: Identify the congruence criterion

The Angle - Side - Angle (ASA) congruence criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Step2: Analyze the given information

We are given that \(\angle RTS\cong\angle TRU\) (angle 1), \(\angle SRT\cong\angle RTU\) (angle 2), and \(\overline{RT}\cong\overline{RT}\) (common side, included side).

Step3: Apply the ASA criterion

Since we have two pairs of congruent angles (\(\angle RTS\cong\angle TRU\) and \(\angle SRT\cong\angle RTU\)) and the included side \(\overline{RT}\) is congruent to itself (\(\overline{RT}\cong\overline{RT}\)), by the ASA (Angle - Side - Angle) congruence criterion, \(\triangle RTU\cong\triangle TRS\).

Answer:

The reason for \(\triangle RTU\cong\triangle TRS\) is the ASA (Angle - Side - Angle) congruence criterion.