Question

given: rs bisects ∠mrq, ∠rms ≅ ∠rqs
which relationship in the diagram is true?
δmnr ≅ δmns by asa
δrms ≅ δrqs by aas
δsnq ≅ δsnm by sss
δqnr ≅ δmnr by hl

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 triangle - congruence criteria

• ASA (Angle - Side - Angle): Two angles and the included side of one triangle are equal to two angles and the included side of another triangle.
• AAS (Angle - Angle - Side): Two angles and a non - included side of one triangle are equal to two angles and the corresponding non - included side of another triangle.
• SSS (Side - Side - Side): All three sides of one triangle are equal to all three sides of another triangle.
• HL (Hypotenuse - Leg): For right - angled triangles, the hypotenuse and one leg of one right - angled triangle are equal to the hypotenuse and one leg of another right - angled triangle.

For \( \triangle RMS \) and \( \triangle RQS \):
We have \( \angle RMS\cong\angle RQS \) (given), \( \angle MRN=\angle QRN \) (because \( RS \) bisects \( \angle MRQ \)), and \( RS = RS \) (common side). This satisfies the AAS (Angle - Angle - Side) congruence criterion.

For \( \triangle MNR\) and \( \triangle MNS\): There is no information about the sides \( NR\) and \( NS\) being equal (required for ASA as we know \( \angle MRN=\angle QRN\) and \( MN = MN\) but no information about the included side for the angles at \( N\)).

For \( \triangle SNQ\) and \( \triangle SNM\): There is no information about \( NQ = NM\) (required for SSS as \( SN=SN\) and \( SQ = SM\) is not given).

For \( \triangle QNR\) and \( \triangle MNR\): There is no information that these are right - angled triangles (required for HL).

Answer:

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