Question

given: \\( \overline{rs} \\) bisects \\( \angle mrq; \angle rms \cong \angle rqs \\)
which relationship in the diagram is true?
\\( \triangle qnr \cong \triangle mnr \\) by hl
\\( \triangle snq \cong \triangle snm \\) by sss
\\( \triangle rms \cong \triangle rqs \\) by aas
\\( \triangle mnr \cong \triangle mns \\) by asa

Explanation:

Step1: Analyze the given information

We know that \( \overline{RS} \) bisects \( \angle MRQ \), so \( \angle MRN=\angle QRN \). Also, \( \angle RMS\cong\angle RQS \). And \( RS = RS \) (common side).

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

The AAS congruence criterion states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.
For \( \triangle RMS \) and \( \triangle RQS \):

• \( \angle RMS\cong\angle RQS \) (given)
• \( \angle MRN=\angle QRN \) (because \( \overline{RS} \) bisects \( \angle MRQ \))
• \( RS = RS \) (common side)

Step3: Check other options

• For \( \triangle QNR\cong\triangle MNR \) by \( HL \): We don't know if these are right - angled triangles (HL is for right - angled triangles: hypotenuse - leg).
• For \( \triangle SNQ\cong\triangle SNM \) by \( SSS \): We don't have information about the equality of all three sides.
• For \( \triangle MNR\cong\triangle MNS \) by \( ASA \): We don't have the necessary angle - side - angle information.

Answer:

\( \triangle RMS\cong\triangle RQS \) by \( AAS \)