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

Explanation:

Identify the target triangles and given components

We want to prove \(\Delta MNS \cong \Delta QNS\) using the SAS (Side-Angle-Side) Congruence Postulate.
From the proof text, we already have:

1. Side: \(\overline{MN} \cong \overline{QN}\) (using Isosceles Triangle Definition).
2. Angle: \(\angle NMS \cong \angle NQS\) (by the Isosceles Triangle Theorem).

Determine the required side for SAS

To use SAS Congruence Postulate for \(\Delta MNS\) and \(\Delta QNS\):

• The congruent angles are \(\angle NMS\) and \(\angle NQS\).
• The sides forming these angles are:
• For \(\angle NMS\): sides \(\overline{MN}\) and \(\overline{MS}\).
• For \(\angle NQS\): sides \(\overline{QN}\) and \(\overline{QS}\).
• Since we already have \(\overline{MN} \cong \overline{QN}\), we need the other pair of sides forming the angles to be congruent: \(\overline{MS} \cong \overline{QS}\).

Apply the definition of bisector

It is given that \(\overline{NR}\) and \(\overline{MQ}\) bisect each other at \(S\).
By the definition of a segment bisector, the bisector divides a segment into two congruent segments.
Since \(\overline{NR}\) bisects \(\overline{MQ}\) at \(S\), point \(S\) is the midpoint of \(\overline{MQ}\).
Therefore, segments \(MS\) and \(QS\) are congruent.

Answer:

• NS and QS
• NS and RS
• MS and RS
• MS and QS (Correct answer)