Question

given: \\( \overline{tr} \cong \overline{ts} \\), \\( \overline{at} \\) bisects \\( \angle rts \\).
prove: \\( \triangle rua \cong \triangle sua \\)

1. \\( \angle rut \cong \angle sut \\)
2. \\( \angle rut \\) and \\( \angle rua \\) are supplementary; \\( \angle sut \\) and \\( \angle sua \\) are supplementary.
3. click here to insert

6.

1. if two angles are a linear pair, then they are supplementary.
2. if two angles are supplementary to congruent angles, then they are congruent.

Explanation:

Step1: Given information

We know that \(\overline{TR}\cong\overline{TS}\) and \(\overline{AT}\) bisects \(\angle RTS\). So, \(\angle RTA=\angle STA\) (definition of angle - bisector). Also, \(AU = AU\) (common side).

Step2: Prove \(\triangle RTA\cong\triangle STA\)

By the Side - Angle - Side (SAS) congruence criterion (\(\overline{TR}\cong\overline{TS}\), \(\angle RTA=\angle STA\), \(\overline{TA}=\overline{TA}\)), \(\triangle RTA\cong\triangle STA\). Then \(RU = SU\) (corresponding parts of congruent triangles are congruent).

Step3: Use angle relationships

Since \(\angle RUT\cong\angle SUT\) (given in step 6 of the proof process) and \(\angle RUT+\angle RUA = 180^{\circ}\), \(\angle SUT+\angle SUA=180^{\circ}\) (step 7, linear - pair supplementary angles). And because \(\angle RUT\cong\angle SUT\), we can say \(\angle RUA\cong\angle SUA\) (if two angles are supplementary to congruent angles, then they are congruent, step 8).

Step4: Prove \(\triangle RUA\cong\triangle SUA\)

Now, in \(\triangle RUA\) and \(\triangle SUA\), we have \(\overline{RU}\cong\overline{SU}\) (from \(\triangle RTA\cong\triangle STA\)), \(\angle RUA\cong\angle SUA\) (from step 3), and \(\overline{AU}\cong\overline{AU}\) (common side). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle RUA\cong\triangle SUA\).

Answer:

\(\triangle RUA\cong\triangle SUA\) is proved using the SAS congruence criterion.