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The problem states that BD bisects ∠ABC, which means that:

\[
\text{m∠ABD} = \text{m∠CBD}
\]

We are also given two expressions for the angles:
- \(\text{m∠ABD} = 5x - 1\)
- \(\text{m∠CBD} = 2x + 5\)

Since BD bisects ∠ABC, these two angles must be equal:

\[
5x - 1 = 2x + 5
\]

### Step 1: Solve for \(x\)
Start by solving the equation \(5x - 1 = 2x + 5\).

1. Subtract \(2x\) from both sides:
   \[
   5x - 2x - 1 = 5
   \]
   \[
   3x - 1 = 5
   \]

2. Add 1 to both sides:
   \[
   3x = 6
   \]

3. Divide by 3:
   \[
   x = 2
   \]

### Step 2: Find the measure of ∠ABC
Since BD bisects ∠ABC, the measure of ∠ABC is the sum of the two equal angles ∠ABD and ∠CBD.

Now, substitute \(x = 2\) into the expression for ∠ABD and ∠CBD:
- \(\text{m∠ABD} = 5(2) - 1 = 10 - 1 = 9^\circ\)
- \(\text{m∠CBD} = 2(2) + 5 = 4 + 5 = 9^\circ\)

Thus, \(\text{m∠ABC} = \text{m∠ABD} + \text{m∠CBD} = 9^\circ + 9^\circ = 18^\circ\).

### Final Answer:
- \(x = 2\)
- \(\text{m∠ABC} = 18^\circ\)

Would you like more details on the steps? 

Here are 5 related questions:
1. How do you use angle bisectors to solve geometric problems?
2. What are some common properties of bisected angles in triangles?
3. Can this method be applied to non-bisected angles?
4. How would you solve a similar problem if it involved supplementary angles?
5. What happens if an exterior angle is bisected in a triangle?

**Tip:** Always check whether the sum of smaller angles equals the larger angle, especially when working with bisectors in geometry.

BD bisects ∠ABC. Solve for x and find m∠ABC. m∠ABD = 5x - 1, m∠CBD = 2x + 5.

Solve for x and find m∠ABC using the Angle Bisector Theorem. Given m∠ABD = 5x - 1 and m∠CBD = 2x + 5, this problem uses algebra and geometry concepts to find the measures of angles. Ideal for Grades 8-10.

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