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Let's go through the problem step by step based on the given pyramid structure. Here's how we can address each part:

### a) **Angle between line [AE] and the plane face ABCD**:
- Since the vertex \(E\) is directly above the center \(D\) of the square base \(ABCD\), the line \(AE\) is perpendicular to the base plane \(ABCD\). Therefore, the angle between \(AE\) and the plane \(ABCD\) is **90°**.

### b) **Angle between the plane face BCE and the face ABCD**:
- To find the angle between the two planes, we need to use the vector normal to each plane.
- Plane \(BCE\): The vectors to consider are from \(B\) to \(C\) and from \(B\) to \(E\).
- Plane \(ABCD\): Since it's a horizontal plane, the normal vector is vertical (i.e., along line \(DE\)).
- The angle between the planes can be calculated using the dot product of these normal vectors.

### c) **Angle between the lines [AE] and [EC]**:
- Using the triangle \(AEC\), we can apply the cosine rule or simple trigonometry to calculate the angle.
- The length of \(AC\) can be calculated using the Pythagorean theorem, since \(ABCD\) is a square with side length 16 cm.

### d) **Find the volume of the pyramid**:
- The formula for the volume of a pyramid is:
  \[
  V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
  \]
  - Base area: \(ABCD\) is a square with side length 16 cm, so the area is \(16^2 = 256 \, \text{cm}^2\).
  - Height: This is the vertical distance from \(E\) to the base, which can be determined using Pythagoras' theorem in triangle \(AED\).
  \[
  h = \sqrt{AE^2 - AD^2}
  \]

### e) **Find the total surface area of the pyramid**:
- The surface area is the sum of the area of the base and the areas of the four triangular faces.
  - Base area = \(256 \, \text{cm}^2\).
  - For each triangular face, the area can be calculated using the formula:
    \[
    A = \frac{1}{2} \times \text{Base} \times \text{Slant Height}
    \]
    - Use \(AB = 16 \, \text{cm}\) and the slant height for the side triangles.

Would you like to go into further detail for any part of this solution? Here's a set of follow-up questions:

1. How do we apply the dot product to find the angle between two planes?
2. Can you show the steps for calculating the slant height in triangle BCE?
3. How do we apply the cosine rule in triangle AEC?
4. How is the height of the pyramid derived from the triangle AED?
5. What is the exact method for calculating the total surface area, including the lateral faces?

**Tip**: Always visualize the geometry in 3D when solving pyramid-related problems to understand angles and dimensions better.

ABCDE is a square-based pyramid. The vertex E is directly above the center of the face ABCD. AB = 16 cm and AE = 20 cm. Calculate the following: a) The angle between line AE and the plane face ABCD. b) The angle between plane BCE and face ABCD. c) The angle between lines AE and EC. d) The volume of the pyramid. e) The total surface area of the pyramid.

This problem involves a square-based pyramid where AB = 16 cm and AE = 20 cm. Learn how to calculate the angles between line AE and the base ABCD, the planes BCE and ABCD, the volume, and the total surface area of the pyramid using trigonometry and geometry.

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