\( \angle E F G \) and \( \angle L M N \) are supplementary angles, \( m \angle E F G=(3 x+17)^{\circ} \), and \( m \angle L M N=\left(\frac{1}{2} x-5\right)^{\circ} \). \( m \angle E F G=\square^{\circ} \) \( m \angle L M N=\square^{\circ} \)

To find the measures of the angles, we start by using the property that supplementary angles add up to \(180^\circ\). Setting up the equation, we have: \[ m \angle EFG + m \angle LMN = 180 \] Substituting in the expressions for the angles: \[ (3x + 17) + \left(\frac{1}{2}x - 5\right) = 180 \] Combining like terms gives us: \[ 3x + \frac{1}{2}x + 17 - 5 = 180 \] \[ \frac{6x}{2} + \frac{1x}{2} + 12 = 180 \] \[ \frac{7x}{2} + 12 = 180 \] Subtracting 12 from both sides: \[ \frac{7x}{2} = 168 \] Multiplying both sides by 2 to eliminate the fraction: \[ 7x = 336 \] Now, divide by 7 to solve for \(x\): \[ x = 48 \] Now that we have \(x\), we can find each angle's measure. For \(m \angle EFG\): \[ m \angle EFG = 3(48) + 17 = 144 + 17 = 161^\circ \] For \(m \angle LMN\): \[ m \angle LMN = \frac{1}{2}(48) - 5 = 24 - 5 = 19^\circ \] Thus, we have: \( m \angle EFG = 161^{\circ} \) \( m \angle LMN = 19^{\circ} \)