Question

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

Explanation:

Step1: Calculate \(m\angle HLM\)

Since \(m\angle KLM = 180^{\circ}\) and \(m\angle KLH=120^{\circ}\), then \(m\angle HLM=m\angle KLM - m\angle KLH\).
\(m\angle HLM = 180^{\circ}-120^{\circ}=60^{\circ}\)

Step2: Calculate \(m\angle HLI\) and \(m\angle ILM\)

We know \(m\angle HLI = 30^{\circ}\) and \(m\angle ILM=15^{\circ}+15^{\circ}=30^{\circ}\) (because \(m\angle ILJ = 15^{\circ}\) and if we assume no other information about bisection for wrong - option elimination, but for the last option: \(m\angle HLI = 30^{\circ}\), \(m\angle ILM=30^{\circ}\))

Step3: Analyze each option

• Option1: \(\angle HLM = 60^{\circ}\), if \(\overrightarrow{LJ}\) bisects \(\angle HLM\), then each part should be \(30^{\circ}\), but \(m\angle MLJ=15^{\circ}+15^{\circ}=30^{\circ}\) and \(m\angle HLJ = 30^{\circ}+15^{\circ}+15^{\circ}=60^{\circ}\), so this is wrong.
• Option2: \(\angle GLJ=60^{\circ}+30^{\circ}+15^{\circ}+15^{\circ}=120^{\circ}\), \(m\angle GLH = 60^{\circ}\), \(m\angle HLJ=60^{\circ}\), so \(\overrightarrow{LH}\) bisects \(\angle GLJ\) (since \(m\angle GLH=m\angle HLJ = 60^{\circ}\))
• Option3: \(m\angle KLG = 60^{\circ}\), \(m\angle HLJ=30^{\circ}+15^{\circ}+15^{\circ}=60^{\circ}\)
• Option4: \(m\angle HLI = 30^{\circ}\), \(m\angle ILM=30^{\circ}\)

Answer:

\(\angle GLJ\) is bisected by \(\overrightarrow{LH}\)