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This problem involves a triangle with perpendicular lines. Here's how we can analyze it step by step:

### Given Information:
- \( \triangle ABC \) is a triangle.
- Line segment \( AD \) is perpendicular to line segment \( BC \), which makes \( \angle ADB = 90^\circ \) (since perpendicular lines form right angles).
- The length of \( BC \) is split into two segments:
  - \( BD = 12n \)
  - \( DC = n^2 + 81 \)
- We are tasked with finding the measure of angle \( \angle ABC \).

### Strategy:
Since we know that \( AD \perp BC \), we can treat \( \triangle ABD \) and \( \triangle ADC \) as right triangles. To determine the value of \( \angle ABC \), we can explore the geometric relationships using trigonometry or potentially a method involving the values for \( n \) and segment lengths.

1. **Calculate the total length of \( BC \)**:
   \[
   BC = BD + DC = 12n + (n^2 + 81) = n^2 + 12n + 81
   \]

2. **Use the properties of right triangles**:
   We may need to apply the Pythagorean theorem or trigonometric functions to calculate angle \( \angle ABC \) once we identify the relationship between sides of the triangle.

Do you know the value of \( n \), or would you like to solve for it first?

Would you like me to continue solving this problem step by step? If so, please provide any additional information if necessary.

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### Follow-Up Questions:
1. What is the value of \( n \), or do you need assistance solving for it?
2. Would you like to explore the trigonometric approach for calculating \( \angle ABC \)?
3. Would you prefer a geometric explanation of perpendicular bisectors and angle bisectors?
4. Should we apply any additional theorems, such as the Pythagorean Theorem?
5. Would you like more examples of problems involving right triangles and perpendicular lines?

#### Tip:
Always check for special right triangles or relationships between the sides of a triangle when dealing with perpendicular lines.

In triangle ABC, AD is perpendicular to BC, with BD = 12n, DC = n^2 + 81, find the angle ∠ABC.

Learn how to find the angle ∠ABC in a right triangle where AD is perpendicular to BC, using geometry and trigonometry. The problem includes right triangle properties and angle calculations, suitable for students in Grades 10-12.

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