Question 1169789
Absolutely, let's break down this pyramid geometry problem.

**Understanding the Problem**

We have a right pyramid with a square base. We need to find specific angles related to the pyramid's edges and faces.

**Given Information**

* Base ABCD is a square with side length AB = 8 cm.
* Height of the pyramid EO = 5 cm (where O is the center of the square base).

**Diagram**

It's helpful to visualize the pyramid.

1.  Draw a square ABCD.
2.  Mark the center of the square as point O.
3.  Draw a line segment EO perpendicular to the square base, with EO = 5 cm.
4.  Connect point E to each vertex of the square (A, B, C, D) to form the pyramid.

**Solutions**

**(a) Angle EAB**

1.  **Triangle EOA:** Triangle EOA is a right triangle, with ∠EOA = 90°.
2.  **OA:** Since O is the center of the square, OA is half the length of the diagonal AC.
    * AC = √(AB² + BC²) = √(8² + 8²) = √(128) = 8√2 cm.
    * OA = (8√2) / 2 = 4√2 cm.
3.  **Tangent:** We can use the tangent function to find angle EAB (let's call it α).
    * tan(α) = EO / OA = 5 / (4√2)
    * α = arctan(5 / (4√2)) ≈ arctan(5 / 5.6568) ≈ arctan(0.884)
    * α ≈ 41.52°

Therefore, angle EAB ≈ 41.52°.

**(b) Angle β between a slant edge and the plane on the base.**

1.  **Slant Edge EB:** We need to find the angle between the slant edge EB and the base.
2.  **Triangle EOB:** Triangle EOB is a right triangle, with ∠EOB = 90°.
3.  **OB:** OB is half the length of the diagonal BD, which is equal to AC.
    * OB = OA = 4√2 cm.
4.  **Tangent:** We can use the tangent function to find angle EBO (β).
    * tan(β) = EO / OB = 5 / (4√2)
    * β = arctan(5 / (4√2)) ≈ 41.52°

Therefore, angle β ≈ 41.52°.

**(c) Angle θ between a slant face and the plane on the base.**

1.  **Slant Face EBC:** We need to find the angle between the slant face EBC and the base ABCD.
2.  **Midpoint of BC:** Let M be the midpoint of BC.
3.  **Triangle EOM:** Triangle EOM is a right triangle, with ∠EOM = 90°.
4.  **OM:** OM is half the length of AB (or CD), so OM = 8 / 2 = 4 cm.
5.  **Tangent:** We can use the tangent function to find angle EMO (θ).
    * tan(θ) = EO / OM = 5 / 4 = 1.25
    * θ = arctan(1.25) ≈ 51.34°

Therefore, angle θ ≈ 51.34°.

**Final Answers**

* **(a) Angle EAB ≈ 41.52°**
* **(b) Angle β ≈ 41.52°**
* **(c) Angle θ ≈ 51.34°**