1. **State the problem:** We need to find the length of the edge $AG$ in the cuboid, given angles $27^\circ$ and $42^\circ$ at vertex $A$, and the length $AD = 63$ mm. 2. **Analyze the geometry:** The angles $27^\circ$ and $42^\circ$ are between edges $AB$, $AD$, and $AG$. Since $AG$ is a diagonal on the face $ADGH$, we can use trigonometry to find its length. 3. **Calculate the length of $AG$:** The angle between $AD$ and $AG$ is $42^\circ$. Using the cosine rule in triangle $A D G$: $$AG = \frac{AD}{\cos 42^\circ}$$ Calculate: $$AG = \frac{63}{\cos 42^\circ} = \frac{63}{0.7431} \approx 84.77 \text{ mm}$$ 4. **Final answer:** The length of $AG$ is approximately $84.77$ mm to 2 decimal places.