Question

raj correctly determined that ray lh is the bisector of ∠gli. which information could he have used to determine this? ( mangle klm = 5mangle ilm ) ( angle glhcongangle ilm ) ( mangle gli = 2mangle glh ) ( mangle gli=\frac{1}{2}mangle glh+\frac{1}{2}mangle hli )

Explanation:

Step1: Recall the definition of an angle bisector

An angle bisector divides an angle into two equal - measure angles. If \(LH\) is the bisector of \(\angle GLI\), then \(\angle GLH=\angle HLI\) and \(m\angle GLI = m\angle GLH+m\angle HLI\). Since \(\angle GLH=\angle HLI\), we have \(m\angle GLI=2m\angle GLH\)

Step2: Analyze each option

• For \(m\angle KLM = 5m\angle ILM\): This equation relates \(\angle KLM\) and \(\angle ILM\), which has no direct relation to \(LH\) being the bisector of \(\angle GLI\)
• For \(\angle GLH\cong\angle ILM\): Congruence of \(\angle GLH\) and \(\angle ILM\) does not imply that \(LH\) bisects \(\angle GLI\)
• For \(m\angle GLI = 2m\angle GLH\): By the definition of an angle bisector (if \(LH\) bisects \(\angle GLI\) into two equal angles \(\angle GLH\) and \(\angle HLI\), then \(m\angle GLI=m\angle GLH + m\angle HLI\) and since \(m\angle GLH=m\angle HLI\), \(m\angle GLI = 2m\angle GLH\))
• For \(m\angle GLI=\frac{1}{2}m\angle GLH+\frac{1}{2}m\angle HLI\): This is an incorrect formula. If \(LH\) bisects \(\angle GLI\), \(m\angle GLI=m\angle GLH + m\angle HLI\) and \(m\angle GLH=m\angle HLI\)

Answer:

\(m\angle GLI = 2m\angle GLH\)