The problem you provided asks for the distance from a point on one of the faces of a dihedral angle to the edge of the angle, given that the dihedral angle between the faces is 45° and the distance from the point to the other face is $5 \sqrt{2}$ cm.

Here’s how to approach solving this problem based on the method shown in the photograph you uploaded:

Given:

• The dihedral angle between the faces is $\angle A = 45^\circ$.
• The distance from the point on the first face to the second face is $AE = 5 \sqrt{2}$ cm.

Goal:
Find the distance from the given point (point E) to the edge of the angle (line $\overline{AD}$).

Solution Steps:

1. Analyze the Geometry of the Situation: The problem likely involves a three-dimensional geometry setup, where there are two intersecting planes. We need to project the point E onto the line of intersection (edge) of these planes.

2. Use the Known Formula for the Distance: Using the projection of the point onto the plane, we can solve for the distance from the point E to the edge of the dihedral angle. The relationship involving the sine of the angle between the planes is crucial for calculating the perpendicular distance.

Following the method provided in the image:

1. Step 1: Analyze the geometry of triangle $ADE$:

   • From the given dihedral angle $\angle A = 45^\circ$, you calculate the perpendicular distance using the sine function.

2. Step 2: Apply trigonometry:

   • Using the sine rule: $AE = \frac{\text{Distance}}{\sin 45^\circ}$ Knowing that $\sin 45^\circ = \frac{\sqrt{2}}{2}$, the calculation is made as shown in the steps.

3. Step 3: Solve the Equation: $\frac{5 \sqrt{2}}{2} = AE$

   Simplifying: $AE = 5 \text{ cm}$

The answer is the distance from point $E$ to the edge of the angle, which is $5$ cm.

If you have any further questions or need more details about these steps, feel free to ask!

Here are 5 related questions you may find interesting:

1. How would you apply these principles to a different angle between the planes, say 60°?
2. Can the formula used in this problem be applied to non-right-angled triangles?
3. What geometric principles underlie the calculation of distances in three-dimensional geometry?
4. How do different types of dihedral angles (acute, right, obtuse) affect the distance calculation?
5. What tools or software could be useful for visualizing and solving dihedral angle problems?

Tip:
Always visualize the geometry of the problem first. Drawing the planes and edge intersection helps significantly in understanding and solving these types of problems!