Question

ray 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 = \frac{1}{2}m∠glh + \frac{1}{2}m∠hli

Explanation:

Step1: Recall the angle - bisector definition

An angle bisector divides an angle into two equal - measure angles. If \(LH\) is the bisector of \(\angle GLJ\), then \(m\angle GLH=m\angle HLJ\) and \(m\angle GLJ = m\angle GLH+m\angle HLJ\).

Step2: Analyze each option

• Option 1: \(\angle GLH\cong\angle ILM\)

This does not relate to the relationship between \(\angle GLH\) and \(\angle HLJ\) (or \(\angle GLJ\)). So, this option is incorrect.

• Option 2: \(m\angle KLM = 5m\angle ILM\)

This equation involves \(\angle KLM\) and \(\angle ILM\), which are not relevant to the bisecting of \(\angle GLJ\) by \(LH\). So, this option is incorrect.

• Option 3: \(m\angle GLJ=2m\angle GLH\)

Let \(m\angle GLH = x\). If \(m\angle GLJ = 2x\), then \(m\angle HLJ=m\angle GLJ - m\angle GLH=2x - x=x\). So, \(m\angle GLH=m\angle HLJ\), which means \(LH\) bisects \(\angle GLJ\) (by the definition of an angle bisector: an angle bisector divides an angle into two angles of equal measure).

• Option 4: \(m\angle GLJ=\frac{1}{2}m\angle GLH+\frac{1}{2}m\angle HLJ\)

If we assume \(m\angle GLH = a\) and \(m\angle HLJ = b\), then \(m\angle GLJ=a + b\). The given equation \(a + b=\frac{1}{2}a+\frac{1}{2}b\) simplifies to \(2a + 2b=a + b\) or \(a + b = 0\) (which is not possible for non - zero angles in this geometric context). So, this option is incorrect.

Answer:

\(m\angle GLJ = 2m\angle GLH\) (the third option)