Question

given that ( mangle klh = 120^{circ} ) and ( mangle klm = 180^{circ} ), which statement about the figure must be true?
( angle glj ) is bisected by ( overrightarrow{lh} ).
( mangle klg = mangle hlj )
( mangle hli = mangle ilm )
( angle hlm ) is bisected by ( overrightarrow{lj} ).

Explanation:

Step1: Analyze each option

• Option 1: \(m\angle HLI=m\angle ILM\)
• There is no information given in the problem to suggest that \(\angle HLI\) and \(\angle ILM\) are equal.
• Option 2: \(m\angle KLG = m\angle HLJ\)
• Given \(m\angle KLH = 120^{\circ}\), then \(m\angle GLH=120^{\circ}- 60^{\circ}=60^{\circ}\). And \(m\angle HLJ = 30^{\circ}+15^{\circ}=45^{\circ}\), \(m\angle KLG = 60^{\circ}\). So \(m\angle KLG

eq m\angle HLJ\).

• Option 3: \(\angle GLJ\) is bisected by \(\overrightarrow{LH}\)
• There is no indication from the given angles (\(m\angle GLH = 60^{\circ}\), and we can't assume equal - division of \(\angle GLJ\) based on the given \(30^{\circ}\) and other angles in an un - related part) that \(\overrightarrow{LH}\) bisects \(\angle GLJ\).
• Option 4: \(\angle HLM\) is bisected by \(\overrightarrow{LJ}\)
• Since \(m\angle HLJ=30^{\circ}+15^{\circ} = 45^{\circ}\) and \(m\angle JLM = 15^{\circ}+30^{\circ}=45^{\circ}\) (because \(m\angle KLH = 120^{\circ}\), \(m\angle KLM=180^{\circ}\), so \(m\angle HLM=180 - 120=60^{\circ}\) is wrong, actually \(m\angle HLM=(180 - 60)^{\circ}=120^{\circ}\), and \(m\angle HLJ=m\angle JLM = 45^{\circ}\))

Answer:

\(\angle HLM\) is bisected by \(\overrightarrow{LJ}\)