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To solve the given problem, we have the following information:

- \( BD \) bisects \( \angle ABC \), meaning \( m\angle ABD = m\angle CBD \).
- \( m\angle ABC = (2 - 16x)^\circ \).
- \( m\angle ABD = (2x + 81)^\circ \).

Since \( BD \) bisects \( \angle ABC \), the two angles \( \angle ABD \) and \( \angle CBD \) are equal, and their sum equals \( m\angle ABC \).

### Step-by-step approach:

1. **Set up the equation:**
   \[
   m\angle ABC = m\angle ABD + m\angle CBD
   \]
   Since \( m\angle ABD = m\angle CBD \), we can write:
   \[
   m\angle ABC = 2 \times m\angle ABD
   \]
   Substituting the expressions for \( m\angle ABC \) and \( m\angle ABD \):
   \[
   (2 - 16x) = 2 \times (2x + 81)
   \]

2. **Solve the equation:**
   \[
   (2 - 16x) = 2(2x + 81)
   \]
   Simplifying the right side:
   \[
   2 - 16x = 4x + 162
   \]
   Move all terms involving \( x \) to one side and constants to the other:
   \[
   -16x - 4x = 162 - 2
   \]
   \[
   -20x = 160
   \]
   Solve for \( x \):
   \[
   x = -8
   \]

3. **Find the angle measures:**
   - \( m\angle ABD = 2x + 81 = 2(-8) + 81 = -16 + 81 = 65^\circ \)
   - \( m\angle CBD = m\angle ABD = 65^\circ \)
   - \( m\angle ABC = 2 - 16x = 2 - 16(-8) = 2 + 128 = 130^\circ \)

### Final answers:
- \( m\angle ABD = 65^\circ \)
- \( m\angle CBD = 65^\circ \)
- \( m\angle ABC = 130^\circ \)

Let me know if you would like more details or have any questions!

Here are 5 related questions to expand on the concept:
1. How do we know that the angles \( \angle ABD \) and \( \angle CBD \) are equal?
2. What is the geometric property of an angle bisector in a triangle?
3. How would the solution change if \( BD \) was not a bisector?
4. What happens if the sum of \( \angle ABC \) is greater than 180° in a triangle?
5. Can you find the exterior angle formed by extending \( AB \)?

**Tip:** When working with angle bisectors, always remember that it divides the angle into two equal parts!

BD bisects ∠ABC. Find m∠ABD, m∠CBD, and m∠ABC. Given m∠ABC = (2 − 16x)° and m∠ABD = (2x + 81)°.

Learn how to solve for m∠ABD, m∠CBD, and m∠ABC in a bisected angle problem, given algebraic expressions. This problem involves the Angle Bisector Theorem and basic algebra. Suitable for students in Grades 9-10.

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