Question

if ( gh = gj = 66 ), ( mangle gih = w + 30^{circ} ), and ( mangle gij = 2w ), what is ( mangle gij )?
( mangle gij=square^{circ} )

Explanation:

Step1: Use the Angle Bisector Theorem

Since \(GH = GJ = 66\) and \(IH\perp GH\), \(IJ\perp GJ\), then \(GI\) is the angle bisector of \(\angle H GJ\). So \(m\angle GIH=m\angle GIJ\).

Step2: Set up the equation

Given \(m\angle GIH = w + 30^{\circ}\) and \(m\angle GIJ = 2w\), we have \(w + 30=2w\).
Solve for \(w\):
\(2w-w=30\), so \(w = 30\).

Step3: Find \(m\angle GIJ\)

Substitute \(w = 30\) into \(m\angle GIJ = 2w\). Then \(m\angle GIJ=2\times30^{\circ}=60^{\circ}\).

Answer:

\(60\)