Question

raj correctly determined that ray lh is the bisector of ∠gli. which information could he have used to determine this? ∠glh ≅ ∠ilm m∠klm = 5m∠ilm m∠gli = 2m∠glh m∠gli = 1/2m∠glh + 1/2m∠hli

Explanation:

Step1: Recall angle - bisector definition

An angle - bisector divides an angle into two equal - measure angles. If ray $LH$ is the bisector of $\angle GLI$, then $m\angle GLH=m\angle HLI$ and $m\angle GLI = 2m\angle GLH=2m\angle HLI$.

Step2: Analyze each option

• Option 1: $\angle GLH\cong\angle ILM$ has no relation to $\angle GLI$ being bisected by $LH$.
• Option 2: $m\angle KLM = 5m\angle ILM$ is about the relationship between $\angle KLM$ and $\angle ILM$, not related to the bisection of $\angle GLI$ by $LH$.
• Option 3: If $m\angle GLI=2m\angle GLH$, then $\angle GLH=\frac{1}{2}m\angle GLI$. This means that ray $LH$ divides $\angle GLI$ into two equal - measure angles (since $m\angle GLH = m\angle HLI$ and $m\angle GLI=m\angle GLH + m\angle HLI$), so ray $LH$ is the bisector of $\angle GLI$.
• Option 4: $m\angle GLI=\frac{1}{2}m\angle GLH+\frac{1}{2}m\angle HLI$ is incorrect. It should be $m\angle GLI=m\angle GLH + m\angle HLI$.

Answer:

$m\angle GLI = 2m\angle GLH$