Question

which statement must be true about the diagram?
○ point k is a midpoint of (overline{jl}).
○ ( mangle jkn = \frac{1}{2}mangle jkm )
○ ray km is an angle bisector of ( angle nkl ).
○ ( jk = \frac{1}{2}kl )

Explanation:

1. Analyze "Point K is a midpoint of \(\overline{JL}\)": We don't have information about the lengths of \(JK\) and \(KL\), so we can't say \(K\) is the midpoint. Eliminate this option.
2. Analyze "\(m\angle JKN=\frac{1}{2}m\angle JKM\)": \(m\angle JKN = 58^{\circ}\), \(m\angle JKM=m\angle JKN + m\angle NKM=58^{\circ}+ 61^{\circ}=119^{\circ}\). \(\frac{1}{2}m\angle JKM=\frac{119^{\circ}}{2} = 59.5^{\circ}

eq58^{\circ}\). Eliminate this option.

1. Analyze "Ray \(KM\) is an angle bisector of \(\angle NKL\)": First, find \(m\angle NKL\). Since \(J - K - L\) is a straight line, \(m\angle JKL = 180^{\circ}\), so \(m\angle NKL=180^{\circ}-m\angle JKN = 180 - 58=122^{\circ}\). The angle between \(KN\) and \(KM\) is \(61^{\circ}\), and the angle between \(KM\) and \(KL\) is \(61^{\circ}\). So \(m\angle NKM=m\angle MKL = 61^{\circ}\), which means \(KM\) bisects \(\angle NKL\). This statement is true.
2. Analyze "\(JK=\frac{1}{2}KL\)": We have no information about the lengths of \(JK\) and \(KL\), so we can't say this. Eliminate this option.

Answer:

Ray \(KM\) is an angle bisector of \(\angle NKL\).