Question

find the measure of \\( \angle g k l \\).
\\( m \angle g k l = \square ^ { \circ } \\)

Explanation:

Step1: Use the property of vertical angles

Since \(IJ\parallel GH\), \((7x - 41)^{\circ}\) and \((8x+11)^{\circ}\) are vertical angles. So \(7x-41 = 8x + 11\).
Solve for \(x\):
\(7x-8x=11 + 41\)
\(-x=52\)
\(x=- 52\)

Step2: Calculate \(m\angle GKL\)

Substitute \(x = - 52\) into \((8x + 11)^{\circ}\)
\(m\angle GKL=8\times(-52)+11\)
\(m\angle GKL=-416 + 11\)
\(m\angle GKL=-405\) (This is wrong, we should use the property of alternate - interior angles. Actually, \((7x - 41)^{\circ}\) and \((8x + 11)^{\circ}\) are same - side interior angles. So \(7x-41+8x + 11=180\))

Step3: Correctly solve for \(x\)

\(7x-41+8x + 11=180\)
\(15x-30 = 180\)
\(15x=180 + 30\)
\(15x=210\)
\(x = 14\)

Step4: Calculate \(m\angle GKL\)

Substitute \(x = 14\) into \((8x + 11)^{\circ}\)
\(m\angle GKL=8\times14+11\)
\(m\angle GKL=112 + 11\)
\(m\angle GKL=123^{\circ}\)

Answer:

\(123\)