Question

select the three statements that guarantee (overline{jk}) is the perpendicular bisector to (overline{lm}).
show your work here
(overline{lm} cong overline{jk}) (overline{kj} cong overline{lk})
(overline{kn} cong overline{mn}) (overline{jn} cong overline{ln})
(overline{jn} cong overline{lm}) (overline{jn} cong overline{ln})
(overline{lk} cong overline{jm}) (overline{jm} cong overline{km})

Explanation:

Step1: Recall the property of perpendicular bisector

If a line is the perpendicular bisector of a segment, then any point on the perpendicular bisector is equidistant from the endpoints of the segment. Since \(LN\) is the perpendicular bisector of \(JK\), we have \(LK = LJ\) and \(MK= MJ\).

Step2: Analyze the congruent segments

For the triangles \(\triangle LKN\) and \(\triangle LJN\), \(LK = LJ\) (from the perpendicular - bisector property), \(KN=JN\) (because \(LN\) bisects \(JK\)), and \(LN = LN\) (common side). So, \(\triangle LKN\cong\triangle LJN\) (by SSS - Side - Side - Side congruence criterion). Similarly, for \(\triangle MKN\) and \(\triangle MJN\), \(MK = MJ\), \(KN = JN\), and \(MN=MN\), so \(\triangle MKN\cong\triangle MJN\).

Answer:

\(\overline{LK}\cong\overline{LJ}\), \(\overline{KN}\cong\overline{JN}\), \(\overline{MK}\cong\overline{MJ}\)