the proof that $\delta mns \cong \delta qns$ is shown. select the answer that best completes the proof.

given: $\delta mnq$ is isosceles with base $\overline{mq}$, and $\overline{nr}$ and $\overline{mq}$ bisect each other at $s$.

prove: $\delta mns \cong \delta qns$

we know that $\delta mnq$ is isosceles with base $\overline{mq}$. so, $\overline{mn} \cong \overline{qn}$ by the definition of isosceles triangle. the base angles of the isosceles triangle, $\angle nms$ and $\angle nqs$, are congruent by the isosceles triangle theorem. it is also given that $\overline{nr}$ and $\overline{mq}$ bisect each other at $s$. segments ______ are therefore congruent by the definition of bisector. thus, $\delta mns \cong \delta qns$ by sas.

ns and qs
ns and rs
ms and rs
ms and qs

Question

the proof that $\delta mns \cong \delta qns$ is shown. select the answer that best completes the proof.

given: $\delta mnq$ is isosceles with base $\overline{mq}$, and $\overline{nr}$ and $\overline{mq}$ bisect each other at $s$.

prove: $\delta mns \cong \delta qns$

we know that $\delta mnq$ is isosceles with base $\overline{mq}$. so, $\overline{mn} \cong \overline{qn}$ by the definition of isosceles triangle. the base angles of the isosceles triangle, $\angle nms$ and $\angle nqs$, are congruent by the isosceles triangle theorem. it is also given that $\overline{nr}$ and $\overline{mq}$ bisect each other at $s$. segments ______ are therefore congruent by the definition of bisector. thus, $\delta mns \cong \delta qns$ by sas.

ns and qs
ns and rs
ms and rs
ms and qs

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "direct_cot", "multiple_choice_formatter", "explain_then_answer", "tutor" ], "needs_drawing": false, "concepts_used": [ "Isosceles Triangle Definition", "SAS Congruence Postulate" ], "new_concepts": [ "Segment Bisector Definition" ], "current_concepts": [ "Isosceles Triangle Definition", "SAS Congruence Postulate", "Segment Bisector Definition" ] } ## Identify the target triangles and given components We want to prove $\Delta MNS \cong \Delta QNS$ using the SAS (Side-Angle-Side) Congruence Postulate. From the proof text, we already have: 1. Side: $\overline{MN} \cong \overline{QN}$ (using **Isosceles Triangle Definition**). 2. Angle: $\angle NMS \cong \angle NQS$ (by the Isosceles Triangle Theorem). ## Determine the required side for SAS To use **SAS Congruence Postulate** for $\Delta MNS$ and $\Delta QNS$: - The congruent angles are $\angle NMS$ and $\angle NQS$. - The sides forming these angles are: - For $\angle NMS$: sides $\overline{MN}$ and $\overline{MS}$. - For $\angle NQS$: sides $\overline{QN}$ and $\overline{QS}$. - Since we already have $\overline{MN} \cong \overline{QN}$, we need the other pair of sides forming the angles to be congruent: $\overline{MS} \cong \overline{QS}$. ## Apply the definition of bisector It is given that $\overline{NR}$ and $\overline{MQ}$ bisect each other at $S$. By the definition of a segment bisector, the bisector divides a segment into two congruent segments. Since $\overline{NR}$ bisects $\overline{MQ}$ at $S$, point $S$ is the midpoint of $\overline{MQ}$. Therefore, segments $MS$ and $QS$ are congruent.

NS and QS

NS and RS

MS and RS

MS and QS

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Segment Bisector Definition" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the target triangles and given components

Determine the required side for SAS

Apply the definition of bisector

Answer: