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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?

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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.

Find the angle between the following line segments and the base plane of the figure:
(a) AC
(b) AB

Learn how to find the angles between line segments AC and AB and the base plane of a cylinder. The problem involves applying the Pythagorean theorem and trigonometric principles. Suitable for high school students in Grades 9-12.

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