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Let's break down the given information to solve the problem.

We are told that:
- \( \overline{BD} \) bisects \( \angle ABC \),
- \( m\angle DBC = 79^\circ \),
- \( m\angle ABC = (9x - 4)^\circ \).

Since \( BD \) bisects \( \angle ABC \), it divides \( \angle ABC \) into two equal parts. This means:
\[
m\angle ABC = 2 \times m\angle DBC.
\]
Therefore, we have the equation:
\[
9x - 4 = 2 \times 79.
\]

Now, let's solve for \( x \).

\[
9x - 4 = 158
\]
\[
9x = 158 + 4
\]
\[
9x = 162
\]
\[
x = \frac{162}{9} = 18
\]

Thus, the value of \( x \) is \( \boxed{18} \).

Would you like further details or have any questions?

Here are 5 related questions to expand on this concept:
1. What is the measure of \( \angle ABC \) after solving for \( x \)?
2. How would the problem change if \( BD \) did not bisect \( \angle ABC \)?
3. Can you calculate the sum of the interior angles at point \( B \)?
4. What would the value of \( x \) be if \( m\angle ABC = (10x - 5)^\circ \) and \( m\angle DBC = 85^\circ \)?
5. How does the concept of angle bisectors relate to the incenter of a triangle?

**Tip:** When solving angle bisector problems, remember that the bisected angle is always split into two equal halves, which can help simplify your calculations.

If BD bisects ∠ABC, m∠DBC = 79°, and m∠ABC = (9x − 4)°, find the value of x.

Learn how to solve for x in an angle bisector problem where BD bisects ∠ABC. The angles are given as m∠DBC = 79° and m∠ABC = (9x − 4)°. This problem introduces the angle bisector theorem and algebraic expressions. Suitable for students in Grades 8-10.

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