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The problem gives the following information about the angles:

- \( m\angle ABC = 12x - 110 \)
- \( m\angle ABD = 3x + 40 \)
- \( m\angle DBC = 2x - 10 \)

Since point \( D \) is inside \( \angle ABC \), we know:

\[
m\angle ABC = m\angle ABD + m\angle DBC
\]

This gives us the equation:

\[
12x - 110 = (3x + 40) + (2x - 10)
\]

Let's solve for \( x \):

\[
12x - 110 = 3x + 40 + 2x - 10
\]
\[
12x - 110 = 5x + 30
\]
\[
12x - 5x = 30 + 110
\]
\[
7x = 140
\]
\[
x = 20
\]

Now substitute \( x = 20 \) into the expression for \( m\angle ABC \):

\[
m\angle ABC = 12(20) - 110 = 240 - 110 = 130^\circ
\]

Thus, \( m\angle ABC = 130^\circ \), which corresponds to option D.

Would you like any additional details or explanations? Here are five related questions for further practice:

1. How would the solution change if \( m\angle ABD \) or \( m\angle DBC \) had different expressions?
2. Can you prove the angle addition postulate in other contexts involving triangle angles?
3. If \( \angle ABC \) were part of a triangle, what would the sum of the angles of the triangle be?
4. How can the method of solving linear equations help in geometry problems?
5. Can you generalize this process to handle more complex angle relationships?

**Tip:** When solving for variables in angle equations, always double-check that your relationships follow geometric rules like the angle addition postulate.

Suppose point D is in the interior of ∠ABC, m∠ABC = 12x − 110, m∠ABD = 3x + 40, and m∠DBC = 2x − 10. What is m∠ABC?

Learn how to solve for m∠ABC using linear equations and the angle addition postulate. This problem covers key concepts like algebra and angle relationships, suitable for students in Grades 8-10. Example: m∠ABC = 12x − 110, m∠ABD = 3x + 40, and m∠DBC = 2x − 10.

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