Question

the proof that δmns ≅ δqns is shown. select the answer that best completes the proof.
given: δmnq is isosceles with base mq, and nr and mq bisect each other at s.
prove: δmns ≅ δqns
we know that δmnq is isosceles with base mq. so, mn ≅ qn by the definition of isosceles triangle. the base angles of the isosceles triangle, ∠nms and ∠nqs, are congruent by the isosceles triangle theorem. it is also given that nr and mq bisect each other at s. segments ____ are therefore congruent by the definition of bisector. thus, δmns ≅ δqns by sas.
ns and qs
ns and rs
ms and rs
ms and qs

Explanation:

Step1: Recall the definition of bisector

If two - line segments bisect each other, the intersection point divides each segment into two equal - length sub - segments. Given that $\overline{NR}$ and $\overline{MQ}$ bisect each other at $S$. For segment $\overline{MQ}$, this means $MS = QS$.

Step2: Recall SAS congruence criterion

The Side - Angle - Side (SAS) congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. We already know that $\angle NMS\cong\angle NQS$ (base angles of isosceles $\triangle MNQ$) and $\overline{MN}\cong\overline{QN}$ (by definition of isosceles triangle). The additional pair of congruent sides needed for SAS is $\overline{MS}$ and $\overline{QS}$.

Answer:

MS and QS