Question

1. in \\( \triangle m n p \\), it is given that \\( \overline{n m} \\) is congruent to \\( \overline{n p} \\). which of the following must be true? (1) \\( \angle p n m \cong \angle p m n \\) (2) point \\( n \\) lies on the perpendicular bisector of \\( \overline{m p} \\) (3) \\( \angle m p n \cong \angle m n p \\) (4) point \\( m \\) lies on the perpendicular bisector of \\( \overline{n p} \\)

Explanation:

Step1: Recall the property of perpendicular bisector

A point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints of the segment.
Since \(NM = NP\) (given that \(\overline{NM}\cong\overline{NP}\)), by the perpendicular bisector theorem, point \(N\) is equidistant from \(M\) and \(P\).

Step2: Analyze each option

• Option (1): \(\angle PNM\) and \(\angle PMN\) are not necessarily congruent. There is no information (like \(MP = NP\) or some angle - side - angle relations) to support this congruence.
• Option (2): Since \(NM = NP\), by the perpendicular bisector theorem (a point is on the perpendicular bisector of a segment if it is equidistant from the endpoints of the segment), point \(N\) lies on the perpendicular bisector of \(\overline{MP}\).
• Option (3): \(\angle MPN\) and \(\angle MNP\) are not necessarily congruent. There is no information (such as \(MP = MN\)) to establish this angle - angle congruence.
• Option (4): There is no information that \(MP = MN\). So, we cannot say that point \(M\) lies on the perpendicular bisector of \(\overline{NP}\).

Answer:

(2) point \(N\) lies on the perpendicular bisector of \(\overline{MP}\)