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We are given that \( BD \) bisects \( \angle ABC \), and the problem asks us to find \( m\angle ABD \), \( m\angle CBD \), and \( m\angle ABC \).

### Step 1: Understand the Geometry
Since \( BD \) bisects \( \angle ABC \), the angles \( \angle ABD \) and \( \angle CBD \) are equal.

We are also given expressions for these angles:
- \( m\angle ABD = (6x + 14)^\circ \)
- \( m\angle CBD = (3x + 29)^\circ \)

Since \( BD \) bisects the angle, we know:
\[
m\angle ABD = m\angle CBD
\]
This allows us to set up the equation:
\[
6x + 14 = 3x + 29
\]

### Step 2: Solve for \( x \)
To solve for \( x \), subtract \( 3x \) from both sides:
\[
6x - 3x + 14 = 29
\]
Simplifying:
\[
3x + 14 = 29
\]
Now, subtract 14 from both sides:
\[
3x = 15
\]
Finally, divide by 3:
\[
x = 5
\]

### Step 3: Calculate the Angles
Now that we have \( x = 5 \), we can substitute this value into the expressions for \( m\angle ABD \) and \( m\angle CBD \).

- \( m\angle ABD = 6(5) + 14 = 30 + 14 = 44^\circ \)
- \( m\angle CBD = 3(5) + 29 = 15 + 29 = 44^\circ \)

Since \( BD \) bisects \( \angle ABC \), we know:
\[
m\angle ABC = m\angle ABD + m\angle CBD = 44^\circ + 44^\circ = 88^\circ
\]

### Final Answers:
- \( m\angle ABD = 44^\circ \)
- \( m\angle CBD = 44^\circ \)
- \( m\angle ABC = 88^\circ \)

Let me know if you need more details, or have any questions!

Here are 5 related questions to explore this topic further:
1. How do you prove that two angles are congruent when a line bisects another angle?
2. What would happen if \( BD \) was not a bisector in this problem? How would the solution change?
3. Can you generalize the method used here to solve similar angle bisector problems?
4. How would you verify if \( BD \) bisects \( \angle ABC \) using geometric tools?
5. What is the importance of the angle sum property in triangle problems?

**Tip:** Always look for relationships like angle bisectors and congruent angles, which can simplify geometry problems significantly!

BD bisects ∠ABC. Find m∠ABD, m∠CBD, and m∠ABC. Given: m∠ABD = (6x + 14)°, m∠CBD = (3x + 29)°.

Learn how to solve a geometry problem where BD bisects ∠ABC. This problem involves solving for m∠ABD, m∠CBD, and m∠ABC using linear equations and angle bisector properties. Suitable for students in Grades 8-10.

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