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": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Geometric Proofs", "Triangle Congruence" ], "new_concepts": [], "current_concepts": [ "Geometric Proofs", "Triangle Congruence", "Segment Bisector" ] } ## Identify the bisected segments $$ \overline{NR} \text{ and } \overline{MQ} \text{ bisect each other at } S \implies S \text{ is the midpoint of both } \overline{NR} \text{ and } \overline{MQ} $$ ## Determine the congruent segments needed for the proof $$ \begin{aligned} &S \text{ is the midpoint of } \overline{MQ} \implies \overline{MS} \cong \overline{QS} \ &\text{In } \triangle MNS \text{ and } \triangle QNS: \ &\overline{MN} \cong \overline{QN} \quad (\text{Side}) \ &\angle NMS \cong \angle NQS \quad (\text{Angle}) \ &\overline{MS} \cong \overline{QS} \quad (\text{Side}) \ &\text{Thus, } \triangle MNS \cong \triangle QNS \text{ by SAS.} \end{aligned} $$ ## Match with the given options $$ \text{The segments that are congruent by the definition of bisector to complete the SAS proof are MS and QS.} $$

NS and QS

NS and RS

MS and RS

MS and QS

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Triangle Congruence" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Identify the bisected segments

Determine the congruent segments needed for the proof

Match with the given options

Answer: