1. **State the problem:** We need to find the measure of angle $\angle GKL$ given two expressions for angles formed by parallel lines and a transversal. 2. **Identify the relationship:** Since lines $EG$ and $IJ$ are parallel and $KL$ is a transversal, the angles $\angle GKL = (8x + 11)^\circ$ and $\angle L = (7x - 41)^\circ$ are either corresponding or alternate interior angles, which are equal. 3. **Set up the equation:** $$8x + 11 = 7x - 41$$ 4. **Solve for $x$:** $$8x + 11 = 7x - 41$$ $$8x - \cancel{7x} + 11 = \cancel{7x} - 41$$ $$x + 11 = -41$$ $$x = -41 - 11$$ $$x = -52$$ 5. **Find $\angle GKL$ by substituting $x$ back:** $$\angle GKL = 8x + 11 = 8(-52) + 11 = -416 + 11 = -405^\circ$$ 6. **Interpret the result:** A negative angle measure suggests an error in angle labeling or interpretation. Since angles cannot be negative, check if the angles are supplementary instead (linear pair). 7. **Check if angles are supplementary:** $$ (8x + 11) + (7x - 41) = 180 $$ $$ 8x + 11 + 7x - 41 = 180 $$ $$ 15x - 30 = 180 $$ $$ 15x = 210 $$ $$ x = 14 $$ 8. **Calculate $\angle GKL$ with $x=14$:** $$ \angle GKL = 8(14) + 11 = 112 + 11 = 123^\circ $$ **Final answer:** $$\boxed{123^\circ}$$