The problem asks to find the angles between specific line segments and the base plane of a cylinder.

1. Angle between line segment AC and the base plane (ABC)
2. Angle between line segment AB and the base plane (ABC)

Explanation of the Setup

• The figure shows a right circular cylinder, where the top of the cylinder is labeled as $A$ and $B$ on the top circular face, and $C$ and $D$ on the base of the cylinder.
• The height of the cylinder is given as 10 cm, and the radius is also 10 cm.

Part (a) Angle between $AC$ and the base plane:

• $AC$ is a slant line from the top of the cylinder (point $A$) to the base at point $C$.
• To calculate this angle, we'll need to understand the geometry of the triangle formed by the vertical height (along the cylinder) and the radius at the base.

Using the Pythagorean theorem:

1. The vertical height from $A$ to the base (along $AD$) is 10 cm.
2. The distance from $A$ to $C$ (horizontal distance) is the radius of the base, which is also 10 cm.

Thus, the length of $AC$ can be calculated using the Pythagorean theorem: $AC = \sqrt{(10^2 + 10^2)} = \sqrt{200} = 10\sqrt{2} \, \text{cm}$

The angle between the segment $AC$ and the base plane is the angle $\theta$ formed by the vertical line and $AC$, given by: $\cos(\theta) = \frac{\text{adjacent (height)}}{\text{hypotenuse (AC)}} = \frac{10}{10\sqrt{2}} = \frac{1}{\sqrt{2}}$ Thus, $\theta = 45^\circ$.

Part (b) Angle between $AB$ and the base plane:

• $AB$ is a line along the top face of the cylinder, and since it lies parallel to the base, the angle between $AB$ and the base plane is simply $0^\circ$, as they are parallel.

Summary of Results:

1. Angle between $AC$ and the base plane: $45^\circ$
2. Angle between $AB$ and the base plane: $0^\circ$

Would you like a more detailed explanation on how these angles were derived or need help with a different part?

Here are 5 related questions for further practice:

1. What is the length of the diagonal from point $A$ to point $D$ on the cylinder?
2. How would the angle change if the height of the cylinder were doubled while keeping the radius constant?
3. Calculate the length of the diagonal from $B$ to $D$.
4. If the cylinder were tilted, how would that affect the angle between $AC$ and the base?
5. What would be the angle between $AB$ and a vertical line from point $A$?

Tip: When solving for angles in three-dimensional figures, always visualize or draw the triangle formed by relevant lines to better understand the relationships between them.