1. **State the problem:** We have a right triangle with a hypotenuse of length 40, an adjacent side to the marked angle of length 15, and we want to find the missing angle at vertex L. 2. **Identify the sides relative to the angle:** The hypotenuse is the longest side (40), the adjacent side to angle L is 15, and the opposite side is unknown. 3. **Choose the appropriate trigonometric ratio:** Since we know the adjacent side and the hypotenuse, the cosine ratio is appropriate. 4. **Recall the cosine formula:** $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ 5. **Substitute the known values:** $$\cos(L) = \frac{15}{40}$$ 6. **Simplify the fraction:** $$\cos(L) = \frac{\cancel{15}}{\cancel{40}} = \frac{3}{8}$$ 7. **Calculate the angle using inverse cosine:** $$L = \cos^{-1}\left(\frac{3}{8}\right)$$ 8. **Evaluate the inverse cosine:** $$L \approx 68.2^\circ$$ **Final answer:** The missing angle at vertex L is approximately $68.2^\circ$.