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:

1. Analyze \(\triangle RMS\) and \(\triangle RQS\):

• Given \(\overline{RS}\) bisects \(\angle MRQ\), so \(\angle RMS\) and \(\angle RQS\) share \(\angle R\) (bisected, so \(\angle MRS=\angle QRS\)).
• Also given \(\angle RMS\cong\angle RQS\), and \(\overline{RS}\) is common to both triangles (\(\overline{RS}\cong\overline{RS}\) by reflexive property).
• For triangle congruence, AAS (Angle - Angle - Side) states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, the triangles are congruent. Here, we have two angles (\(\angle MRS=\angle QRS\), \(\angle RMS=\angle RQS\)) and the non - included side \(\overline{RS}\) congruent, so \(\triangle RMS\cong\triangle RQS\) by AAS.
• For \(\triangle MNR\cong\triangle MNS\) by ASA: We don't have enough information to confirm the angles and sides for ASA (Angle - Side - Angle) congruence.
• For \(\triangle SNQ\cong\triangle SNM\) by SSS: We don't have information about all three sides being congruent.
• For \(\triangle QNR\cong\triangle MNR\) by HL: HL (Hypotenuse - Leg) is for right triangles, and we don't know if these are right triangles or have the hypotenuse - leg relationship.

Answer:

\(\boldsymbol{\triangle RMS \cong \triangle RQS}\) by AAS