The problem involves finding the length of $AG$ in a cuboid where the triangle formed by the lines $AG$, $AB$, and $AE$ are right triangles. We are given the following details:

• The length of $AB = 42 \, \text{mm}$
• Angle $\angle EAB = 27^\circ$
• Angle $\angle AGH = 36^\circ$

Steps to find the length of $AG$:

1. Use trigonometry to break down the sides of the triangles:
   • For triangle $ABE$, we can use the sine or cosine of $27^\circ$ to find the length of $AE$, which will help us use it later in the triangle involving $AG$.
   • We can also use the angle $36^\circ$ to calculate the projection of the line $AG$ using a similar approach.

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