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. options: ns and qs, ns and rs, ms and rs, ms and qs

Explanation:

Step1: Identify base bisector segments

In isosceles $\triangle MNQ$ with base $\overline{NQ}$, the angle bisector of the vertex angle also bisects the base. Since $\overline{NR}$ and $\overline{MQ}$ bisect each other at $S$, the bisected segments are $\overline{NS}$ and $\overline{QS}$.

Step2: Confirm congruent segments

By definition of a bisector, $\overline{NS} \cong \overline{QS}$.

Answer:

NS and QS