Question

the proof that $\triangle mns \cong \triangle qns$ is shown. select the answer that best completes the proof. given: $\triangle mnq$ is isosceles with base $\overline{mq}$, and $\overline{nr}$ and $\overline{mq}$ bisect each other at $s$. prove: $\triangle mns \cong \triangle qns$ diagram of a quadrilateral with vertices $m$, $q$, $n$, $r$ and intersection $s$ of $\overline{nr}$ and $\overline{mq}$ we know that $\triangle mnq$ 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, $\triangle mns \cong \triangle qns$ by sas. \\(\circ\\) $ns$ and $qs$ \\(\circ\\) $ns$ and $rs$ \\(\circ\\) $ms$ and $rs$ \\(\circ\\) $ms$ and $qs$

Explanation:

To prove \(\triangle MNS \cong \triangle QNS\) by SAS, we need two sides and the included angle. We know \(\overline{MN} \cong \overline{QN}\) and \(\angle NMS \cong \angle NQS\). Now, since \(\overline{NR}\) and \(\overline{MQ}\) bisect each other at \(S\), by the definition of a bisector, the segments created by the bisecting point \(S\) on \(\overline{MQ}\) (i.e., \(MS\) and \(QS\)) are congruent. So we need to identify which segments are congruent due to the bisecting. The bisecting of \(\overline{MQ}\) at \(S\) means \(MS = QS\) (since a bisector divides a segment into two equal parts). Let's analyze the options:

• Option 1: \(NS\) and \(QS\) – There's no reason from the bisecting of \(\overline{MQ}\) or \(\overline{NR}\) to conclude these are congruent.
• Option 2: \(NS\) and \(RS\) – This would be from the bisecting of \(\overline{NR}\), but we need a side for the SAS related to \(\overline{MQ}\) here.
• Option 3: \(MS\) and \(RS\) – These are from different segments (\(\overline{MQ}\) and \(\overline{NR}\)), no direct congruence from the given bisecting info for this pair.
• Option 4: \(MS\) and \(QS\) – Since \(\overline{MQ}\) is bisected at \(S\), \(S\) is the midpoint of \(\overline{MQ}\), so \(MS = QS\) by definition of a bisector (midpoint divides the segment into two congruent parts). This gives us the second side needed for SAS (along with \(\overline{MN} \cong \overline{QN}\) and the included angle \(\angle NMS \cong \angle NQS\)).

Answer:

D. MS and QS