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We are given that \( BD \) bisects \( \angle ABC \), and the measurements of the angles are expressed in terms of \( x \). Here's how we can solve for \( m\angle ABD \), \( m\angle CBD \), and \( m\angle ABC \):

### Step-by-Step Solution:

1. **Expression for the angles**:
   - \( m\angle ABD = (3x + 6)^\circ \)
   - \( m\angle CBD = (7x - 18)^\circ \)

2. **Angle bisector property**:
   Since \( BD \) bisects \( \angle ABC \), we know that:
   \[
   m\angle ABD = m\angle CBD
   \]

   Therefore, we can set the two expressions equal to each other:
   \[
   3x + 6 = 7x - 18
   \]

3. **Solve for \( x \)**:
   \[
   3x + 6 = 7x - 18
   \]
   Subtract \( 3x \) from both sides:
   \[
   6 = 4x - 18
   \]
   Add 18 to both sides:
   \[
   24 = 4x
   \]
   Divide by 4:
   \[
   x = 6
   \]

4. **Find \( m\angle ABD \) and \( m\angle CBD \)**:
   Substitute \( x = 6 \) into the expressions for \( m\angle ABD \) and \( m\angle CBD \):
   \[
   m\angle ABD = 3(6) + 6 = 18 + 6 = 24^\circ
   \]
   \[
   m\angle CBD = 7(6) - 18 = 42 - 18 = 24^\circ
   \]

5. **Find \( m\angle ABC \)**:
   Since \( BD \) bisects \( \angle ABC \), we know:
   \[
   m\angle ABC = m\angle ABD + m\angle CBD = 24^\circ + 24^\circ = 48^\circ
   \]

### Final Answers:
- \( m\angle ABD = 24^\circ \)
- \( m\angle CBD = 24^\circ \)
- \( m\angle ABC = 48^\circ \)

Would you like further details or have any other questions?

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Here are 5 related questions:
1. How would the solution change if \( BD \) did not bisect \( \angle ABC \)?
2. What happens if \( BD \) bisects a right angle at \( \angle ABC \)?
3. Can angle bisectors always divide an angle into two equal parts in non-Euclidean geometry?
4. How would you find \( x \) if the bisected angle had been expressed differently?
5. How does the concept of angle bisectors apply in triangle constructions?

**Tip:** When solving angle-related problems, always check if there are additional geometric properties, such as symmetry or bisectors, to simplify the solution.

BD bisects ∠ABC. Find m∠ABD, m∠CBD, and m∠ABC.

Learn how to solve a geometry problem involving an angle bisector. BD bisects ∠ABC, and the angle measures are expressed in terms of x. This step-by-step solution includes solving for m∠ABD, m∠CBD, and m∠ABC using algebra and geometric properties. Suitable for students in Grades 8-10.

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