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The problem asks to solve for \(x\) and find the measure of \(\angle ABC\), given that \(BD\) bisects \(\angle ABC\), meaning \(\angle ABD = \angle CBD\). The expressions for the two angles are:

- \(\angle ABD = 11x - 4\)
- \(\angle CBD = 8x + 5\)

Since \(BD\) bisects \(\angle ABC\), we can set the two expressions equal to each other:

\[
11x - 4 = 8x + 5
\]

### Step 1: Solve for \(x\)
1. Subtract \(8x\) from both sides:
\[
11x - 8x - 4 = 5
\]
\[
3x - 4 = 5
\]

2. Add 4 to both sides:
\[
3x = 9
\]

3. Divide by 3:
\[
x = 3
\]

### Step 2: Find \(\angle ABC\)

Since \(BD\) bisects \(\angle ABC\), we know that \(\angle ABC = 2 \times \angle ABD\) (or \(\angle ABC = 2 \times \angle CBD\), as they are equal).

First, substitute \(x = 3\) into either expression for \(\angle ABD\) or \(\angle CBD\). Let's use \(\angle ABD = 11x - 4\):

\[
\angle ABD = 11(3) - 4 = 33 - 4 = 29^\circ
\]

Since \(\angle ABD = \angle CBD\), \(\angle ABC = 2 \times 29 = 58^\circ\).

### Final Answer:
- \(x = 3\)
- \(\angle ABC = 58^\circ\)

Would you like further details on any of these steps or clarifications?

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Here are 5 related questions to expand your understanding:
1. What happens if \(BD\) does not bisect \(\angle ABC\)?
2. How can we use properties of angle bisectors in triangles?
3. How do we solve if \(\angle ABD\) and \(\angle CBD\) were unequal?
4. What other methods can be used to verify the measure of an angle bisector?
5. How does this problem relate to solving for unknowns in geometry?

**Tip:** When solving for variables in angle relationships, always use given geometrical properties, like bisectors, as they offer critical equalities or proportions.

BD bisects \(\angle ABC\). Solve for x and find \(\angle ABC\). Given: m\(\angle ABD = 11x - 4\) and m\(\angle CBD = 8x + 5\).

This geometry problem involves solving for x and finding the measure of angle ABC, where BD bisects angle ABC. The expressions for the bisected angles are given as m\(\angle ABD = 11x - 4\) and m\(\angle CBD = 8x + 5\). Learn step-by-step how to solve it.

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