Question

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

Explanation:

Step1: 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.

Step2: Analyze the given information

Since \( \overline{RS} \) bisects \( \angle MRQ \), we have \( \angle MRN=\angle QRN \). We are given \( \angle RMS\cong\angle RQS \), and \( RS = RS \) (common side).
In \( \triangle RMS \) and \( \triangle RQS \):

• \( \angle RMS\cong\angle RQS \) (given)
• \( \angle MRN=\angle QRN \) (angle - bisector property)
• \( RS = RS \) (common side)

Step3: Check other options

• For \( \triangle MNR\cong\triangle MNS \) by ASA: There is no information about the sides \( NR = NS \) or the angles \( \angle MNR=\angle MNS \) to apply ASA.
• For \( \triangle SNQ\cong\triangle SNM \) by SSS: There is no information about the sides \( NQ = NM \) to apply SSS.
• For \( \triangle QNR\cong\triangle MNR \) by HL: There is no information indicating that these are right - triangles.

Answer:

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