Question

the proof that ( \triangle mnscong\triangle qns ) is shown. select the answer that best completes the proof.

given: ( \triangle mnq ) is isosceles with base ( overline{mq} ), and ( nr ) and ( mq ) bisect each other at ( s ).
prove: ( \triangle mnscong\triangle qns )

we know that ( \triangle mnq ) is isosceles with base ( overline{mq} ). so, ( overline{mn}congoverline{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 mnscong\triangle qns ) by sas.

ms and qs

ns and qs

ns and rs

ms and rs

Explanation:

Step1: Recall the definition of bisector

If two segments bisect each other, the segments formed by the bisection are congruent. Since \(NR\) and \(MQ\) bisect each other at \(S\), for segment \(MQ\), we have \(MS = QS\) (by the definition of bisector: a bisector of a segment divides the segment into two equal - length parts).

Step2: Check the SAS (Side - Angle - Side) congruence criterion

We know that \(MN\cong QN\) (given that \(\triangle MNQ\) is isosceles with base \(MQ\)), \(\angle NMS\cong\angle NQS\) (base angles of an isosceles triangle \(\triangle MNQ\)), and if we can show a pair of sides adjacent to the congruent angles are congruent. The sides adjacent to \(\angle NMS\) and \(\angle NQS\) are \(MS\) and \(QS\) (from the bisection of \(MQ\) by \(NR\) at \(S\)).

Answer:

MS and QS