Question

gle congruence
given: (overline{rs}) bisects (angle mrq); (angle rmscongangle rqs)
which relationship in the diagram is true?
(\triangle rmscong\triangle rqs) by aas
(\triangle qnrcong\triangle mnr) by hl
(\triangle snqcong\triangle snm) by sss
(\triangle mnrcong\triangle mns) by asa

Explanation:

Step1: Recall triangle congruence theorems

• AAS (Angle - Angle - Side): 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 triangles are congruent.
• HL (Hypotenuse - Leg): For right - triangles, if the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the triangles are congruent. But we don't know if \(\triangle QNR\) and \(\triangle MNR\) are right - triangles from the given information.
• SSS (Side - Side - Side): If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. We don't have information about all three sides of \(\triangle SNQ\) and \(\triangle SNM\).
• ASA (Angle - Side - Angle): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. We don't have information about the included side for \(\triangle MNR\) and \(\triangle MNS\).

Since \(RS\) bisects \(\angle MRQ\), \(\angle MR S=\angle QR S\). We are given \(\angle RMS\cong\angle RQS\), and \(RS = RS\) (common side). So, for \(\triangle RMS\) and \(\triangle RQS\), we have two angles (\(\angle RMS\cong\angle RQS\) and \(\angle MR S=\angle QR S\)) and a non - included side (\(RS\)) that are congruent.

Answer:

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