Question

given \\( \overline { r s } \\) bisects \\( \angle m r q , \angle r m s \cong \angle r q s \\) which relationship in the diagram is true? \\( \delta m n r \cong \delta m n s \\) by asa \\( \delta r m s \cong \delta r q s \\) by aas \\( \delta s n q \cong \delta s n m \\) by sss \\( \delta q n r \cong \delta m n r \\) by hl

Explanation:

Step1: Analyze the given information

We know that $\overline{RS}$ bisects $\angle MRQ$, so $\angle MRS=\angle QRS$. Also, $\angle RMS\cong\angle RQS$ and $RS = RS$ (common side).

Step2: Recall the congruence criteria

The $AAS$ (Angle - Angle - Side) 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.
In $\Delta RMS$ and $\Delta RQS$, we have $\angle RMS=\angle RQS$ (given), $\angle MRS=\angle QRS$ (because $\overline{RS}$ bisects $\angle MRQ$), and $RS = RS$ (common side).

Step3: Check other options

• For $\Delta MNR\cong\Delta MNS$ by $ASA$: There is no information given about the angles and sides to satisfy the $ASA$ criterion.
• For $\Delta SNO\cong\Delta SNM$ by $SSS$: There is no information about the equality of all three sides.
• For $\Delta QNR\cong\Delta MNR$ by $HL$: There is no right - angle information given to use the $HL$ (Hypotenuse - Leg) criterion.

So, the correct relationship is $\Delta RMS\cong\Delta RQS$ by $AAS$.

Answer:

$\Delta RMS\cong\Delta RQS$ by $AAS$